Probabilistic choice
You are encouraged to solve this task according to the task description, using any language you may know.
Given a mapping between items and their required probability of occurrence, generate a million items randomly subject to the given probabilities and compare the target probability of occurrence versus the generated values.
The total of all the probabilities should equal one. (Because floating point arithmetic is involved, this is subject to rounding errors).
Use the following mapping to test your programs:
aleph 1/5.0 beth 1/6.0 gimel 1/7.0 daleth 1/8.0 he 1/9.0 waw 1/10.0 zayin 1/11.0 heth 1759/27720 # adjusted so that probabilities add to 1
- Related task
Ada
with Ada.Numerics.Float_Random; use Ada.Numerics.Float_Random;
with Ada.Text_IO; use Ada.Text_IO;
procedure Random_Distribution is
Trials : constant := 1_000_000;
type Outcome is (Aleph, Beth, Gimel, Daleth, He, Waw, Zayin, Heth);
Pr : constant array (Outcome) of Uniformly_Distributed :=
(1.0/5.0, 1.0/6.0, 1.0/7.0, 1.0/8.0, 1.0/9.0, 1.0/10.0, 1.0/11.0, 1.0);
Samples : array (Outcome) of Natural := (others => 0);
Value : Uniformly_Distributed;
Dice : Generator;
begin
for Try in 1..Trials loop
Value := Random (Dice);
for I in Pr'Range loop
if Value <= Pr (I) then
Samples (I) := Samples (I) + 1;
exit;
else
Value := Value - Pr (I);
end if;
end loop;
end loop;
-- Printing the results
for I in Pr'Range loop
Put (Outcome'Image (I) & Character'Val (9));
Put (Float'Image (Float (Samples (I)) / Float (Trials)) & Character'Val (9));
if I = Heth then
Put_Line (" rest");
else
Put_Line (Uniformly_Distributed'Image (Pr (I)));
end if;
end loop;
end Random_Distribution;
Sample output:
ALEPH 2.00167E-01 2.00000E-01 BETH 1.67212E-01 1.66667E-01 GIMEL 1.42290E-01 1.42857E-01 DALETH 1.24186E-01 1.25000E-01 HE 1.11455E-01 1.11111E-01 WAW 1.00325E-01 1.00000E-01 ZAYIN 9.10220E-02 9.09091E-02 HETH 6.33430E-02 rest
ALGOL 68
INT trials = 1 000 000;
MODE LREAL = LONG REAL;
MODE ITEM = STRUCT(
STRING name,
INT prob count,
LREAL expect,
mapping
);
INT col width = 9;
FORMAT real repr = $g(-col width+1, 6)$,
item repr = $"Name: "g", Prob count: "g(0)", Expect: "f(real repr)", Mapping: ", f(real repr)l$;
[8]ITEM items := (
( "aleph", 0, ~, ~ ),
( "beth", 0, ~, ~ ),
( "gimel", 0, ~, ~ ),
( "daleth", 0, ~, ~ ),
( "he", 0, ~, ~ ),
( "waw", 0, ~, ~ ),
( "zayin", 0, ~, ~ ),
( "heth", 0, ~, ~ )
);
main:
(
LREAL offset = 5; # const #
# initialise items #
LREAL total sum := 0;
FOR i FROM LWB items TO UPB items - 1 DO
expect OF items[i] := 1/(i-1+offset);
total sum +:= expect OF items[i]
OD;
expect OF items[UPB items] := 1 - total sum;
mapping OF items[LWB items] := expect OF items[LWB items];
FOR i FROM LWB items + 1 TO UPB items DO
mapping OF items[i] := mapping OF items[i-1] + expect OF items[i]
OD;
# printf((item repr, items)) #
# perform the sampling #
PROC sample = (REF[]LREAL mapping)INT:(
INT out;
LREAL rand real = random;
FOR j FROM LWB items TO UPB items DO
IF rand real < mapping[j] THEN
out := j;
done
FI
OD;
done: out
);
FOR i TO trials DO
prob count OF items[sample(mapping OF items)] +:= 1
OD;
FORMAT indent = $17k$;
# print the results #
printf(($"Trials: "g(0)l$, trials));
printf(($"Items:"$,indent));
FOR i FROM LWB items TO UPB items DO printf(($gn(col width)k" "$, name OF items[i])) OD;
printf(($l"Target prob.:"$, indent, $f(real repr)" "$, expect OF items));
printf(($l"Attained prob.:"$, indent));
FOR i FROM LWB items TO UPB items DO printf(($f(real repr)" "$, prob count OF items[i]/trials)) OD;
printf($l$)
)Sample output:
Trials: 1000000 Items: aleph beth gimel daleth he waw zayin heth Target prob.: 0.200000 0.166667 0.142857 0.125000 0.111111 0.100000 0.090909 0.063456 Attained prob.: 0.199987 0.166917 0.142531 0.124203 0.111338 0.099702 0.091660 0.063662
AppleScript
AppleScript does have a random number command, but this is located in the StandardAdditions OSAX and invoking it a million times can take quite a while. Since Mac OS X 10.11, it's been possible to use the randomising features of the system's "GameplayKit" framework, which are faster to access.
use AppleScript version "2.5" -- Mac OS X 10.11 (El Capitan) or later.
use framework "Foundation"
use framework "GameplayKit"
on probabilisticChoices(mapping, picks)
script o
property mapping : {}
end script
-- Make versions of the mapping records with additional 'actual' properties …
set mapping to current application's class "NSMutableArray"'s arrayWithArray:(mapping)
tell mapping to makeObjectsPerformSelector:("addEntriesFromDictionary:") withObject:({actual:0})
-- … ensuring that they're sorted (for accuracy) in descending order (for efficiency) of probability.
set descriptor to current application's class "NSSortDescriptor"'s sortDescriptorWithKey:("probability") ascending:(false)
tell mapping to sortUsingDescriptors:({descriptor})
set o's mapping to mapping as list
set rndGenerator to current application's class "GKRandomDistribution"'s distributionForDieWithSideCount:(picks)
set onePickth to 1 / picks
repeat picks times
-- Get a random number between 0.0 and 1.0.
set r to rndGenerator's nextUniform()
-- Interpret the probability of the number occurring in the range it does
-- as picking the item with the same probability of being picked.
repeat with thisRecord in o's mapping
set r to r - (thisRecord's probability)
if (r ≤ 0) then
set thisRecord's actual to (thisRecord's actual) + onePickth
exit repeat
end if
end repeat
end repeat
return o's mapping
end probabilisticChoices
on task()
set mapping to {{|item|:"aleph", probability:1 / 5}, {|item|:"beth", probability:1 / 6}, ¬
{|item|:"gimel", probability:1 / 7}, {|item|:"daleth", probability:1 / 8}, ¬
{|item|:"he", probability:1 / 9}, {|item|:"waw", probability:1 / 10}, ¬
{|item|:"zayin", probability:1 / 11}, {|item|:"heth", probability:1759 / 27720}}
set picks to 1000000
set theResults to probabilisticChoices(mapping, picks)
set output to {}
set template to {"|item|:", missing value, ", probability:", missing value, ", actual:", missing value}
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ""
repeat with thisRecord in theResults
set {|item|:item 2 of template, probability:item 4 of template, actual:item 6 of template} to thisRecord
set {|item|:template's second item, probability:template's fourth item, actual:template's sixth item} to thisRecord
set end of output to template as text
end repeat
set AppleScript's text item delimiters to "}, ¬
{"
set output to "{ ¬
{" & output & "} ¬
}"
set AppleScript's text item delimiters to astid
return output
end task
task()
- Output:
"{ ¬
{|item|:aleph, probability:0.2, actual:0.20033}, ¬
{|item|:beth, probability:0.166666666667, actual:0.166744}, ¬
{|item|:gimel, probability:0.142857142857, actual:0.142403}, ¬
{|item|:daleth, probability:0.125, actual:0.125195}, ¬
{|item|:he, probability:0.111111111111, actual:0.110284}, ¬
{|item|:waw, probability:0.1, actual:0.100505}, ¬
{|item|:zayin, probability:0.090909090909, actual:0.090721}, ¬
{|item|:heth, probability:0.063455988456, actual:0.063817} ¬
}"
Arturo
nofTrials: 10000
probabilities: #[
aleph: to :rational [1 5]
beth: to :rational [1 6]
gimel: to :rational [1 7]
daleth: to :rational [1 8]
he: to :rational [1 9]
waw: to :rational [1 10]
zayin: to :rational [1 11]
heth: to :rational [1759 27720]
]
samples: #[]
loop 1..nofTrials 'x [
z: random 0.0 1.0
loop probabilities [item,prob][
if? z < prob [
unless key? samples item -> samples\[item]: 0
samples\[item]: samples\[item] + 1
break
]
else [
z: z - prob
]
]
]
[s1, s2]: 0.0
print [pad.right "Item" 10 pad "Target" 10 pad "Tesults" 10 pad "Differences" 15]
print repeat "-" 50
loop probabilities [item,prob][
r: samples\[item] // nofTrials
s1: s1 + r*100
s2: s2 + prob*100
print [
pad.right item 10
pad to :string prob 10
pad to :string round.to:4 r 10
pad to :string round.to:4 to :floating 100*1-r//prob 13 "%"
]
]
print repeat "-" 50
print [
pad.right "Total:" 10
pad to :string to :floating s2 10
pad to :string s1 10
]
- Output:
Item Target Tesults Differences -------------------------------------------------- aleph 1/5 0.1954 2.3 % beth 1/6 0.1654 0.76 % gimel 1/7 0.1397 2.21 % daleth 1/8 0.1234 1.28 % he 1/9 0.112 -0.8 % waw 1/10 0.1032 -3.2 % zayin 1/11 0.0946 -4.06 % heth 1759/27720 0.0663 -4.4819 % -------------------------------------------------- Total: 100.0 100.0
AutoHotkey
contributed by Laszlo on the ahk forum
s1 := "aleph", p1 := 1/5.0 ; Input
s2 := "beth", p2 := 1/6.0
s3 := "gimel", p3 := 1/7.0
s4 := "daleth", p4 := 1/8.0
s5 := "he", p5 := 1/9.0
s6 := "waw", p6 := 1/10.0
s7 := "zayin", p7 := 1/11.0
s8 := "heth", p8 := 1-p1-p2-p3-p4-p5-p6-p7
n := 8, r0 := 0, r%n% := 1 ; auxiliary data
Loop % n-1
i := A_Index-1, r%A_Index% := r%i% + p%A_Index% ; cummulative distribution
Loop 1000000 {
Random R, 0, 1.0
Loop %n% ; linear search
If (R < r%A_Index%) {
c%A_Index%++
Break
}
}
; Output
Loop %n%
t .= s%A_Index% "`t" p%A_Index% "`t" c%A_Index%*1.0e-6 "`n"
Msgbox %t%
/*
output:
---------------------------
aleph 0.200000 0.199960
beth 0.166667 0.166146
gimel 0.142857 0.142624
daleth 0.125000 0.124924
he 0.111111 0.111226
waw 0.100000 0.100434
zayin 0.090909 0.091344
heth 0.063456 0.063342
---------------------------
*/
AWK
#!/usr/bin/awk -f
BEGIN {
ITERATIONS = 1000000
delete symbMap
delete probMap
delete counts
initData();
for (i = 0; i < ITERATIONS; i++) {
distribute(rand())
}
showDistributions()
exit
}
function distribute(rnd, cnt, symNum, sym, symPrb) {
cnt = length(symbMap)
for (symNum = 1; symNum <= cnt; symNum++) {
sym = symbMap[symNum];
symPrb = probMap[sym];
rnd -= symPrb;
if (rnd <= 0) {
counts[sym]++
return;
}
}
}
function showDistributions( s, sym, prb, actSum, expSum, totItr) {
actSum = 0.0
expSum = 0.0
totItr = 0
printf "%-7s %-7s %-5s %-5s\n", "symb", "num.", "act.", "expt."
print "------- ------- ----- -----"
for (s = 1; s <= length(symbMap); s++) {
sym = symbMap[s]
prb = counts[sym]/ITERATIONS
actSum += prb
expSum += probMap[sym]
totItr += counts[sym]
printf "%-7s %7d %1.3f %1.3f\n", sym, counts[sym], prb, probMap[sym]
}
print "------- ------- ----- -----"
printf "Totals: %7d %1.3f %1.3f\n", totItr, actSum, expSum
}
function initData( sym) {
srand()
probMap["aleph"] = 1.0 / 5.0
probMap["beth"] = 1.0 / 6.0
probMap["gimel"] = 1.0 / 7.0
probMap["daleth"] = 1.0 / 8.0
probMap["he"] = 1.0 / 9.0
probMap["waw"] = 1.0 / 10.0
probMap["zyin"] = 1.0 / 11.0
probMap["heth"] = 1759.0 / 27720.0
symbMap[1] = "aleph"
symbMap[2] = "beth"
symbMap[3] = "gimel"
symbMap[4] = "daleth"
symbMap[5] = "he"
symbMap[6] = "waw"
symbMap[7] = "zyin"
symbMap[8] = "heth"
for (sym in probMap)
counts[sym] = 0;
}
Example output:
symb num. act. expt. ------- ------- ----- ----- aleph 200598 0.201 0.200 beth 166317 0.166 0.167 gimel 142391 0.142 0.143 daleth 125051 0.125 0.125 he 110658 0.111 0.111 waw 100464 0.100 0.100 zyin 90649 0.091 0.091 heth 63872 0.064 0.063 ------- ------- ----- ----- Totals: 1000000 1.000 1.000
Rounding off makes the results look perfect.
BASIC256
dim letters$ = {"aleph", "beth", "gimel", "daleth", "he", "waw", "zayin", "heth"}
dim actual(8) fill 0 ## all zero by default
dim probs = {1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0, 0}
dim cumProbs(8)
cumProbs[0] = probs[0]
for i = 1 to 6
cumProbs[i] = cumProbs[i - 1] + probs[i]
next i
cumProbs[7] = 1.0
probs[7] = 1.0 - cumProbs[6]
n = 1000000
sum = 0.0
for i = 1 to n
rnd = rand ## random number where 0 <= rand < 1
begin case
case rnd <= cumProbs[0]
actual[0] += 1
case rnd <= cumProbs[1]
actual[1] += 1
case rnd <= cumProbs[2]
actual[2] += 1
case rnd <= cumProbs[3]
actual[3] += 1
case rnd <= cumProbs[4]
actual[4] += 1
case rnd <= cumProbs[5]
actual[5] += 1
case rnd <= cumProbs[6]
actual[6] += 1
else
actual[7] += 1
end case
next i
sumActual = 0
print "Letter", " Actual", "Expected"
print "------", "--------", "--------"
for i = 0 to 7
print ljust(letters$[i],14," ");
print ljust(actual[i]/n,8,"0"); " ";
sumActual += actual[i]/n
print ljust(probs[i],8,"0")
next i
print " ", "--------", "--------"
print " ", ljust(sumActual,8,"0"), "1.000000"
endBBC BASIC
DIM item$(7), prob(7), cnt%(7)
item$() = "aleph","beth","gimel","daleth","he","waw","zayin","heth"
prob() = 1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0, 1759/27720
IF ABS(SUM(prob())-1) > 1E-6 ERROR 100, "Probabilities don't sum to 1"
FOR trial% = 1 TO 1E6
r = RND(1)
p = 0
FOR i% = 0 TO DIM(prob(),1)
p += prob(i%)
IF r < p cnt%(i%) += 1 : EXIT FOR
NEXT
NEXT
@% = &2060A
PRINT "Item actual theoretical"
FOR i% = 0 TO DIM(item$(),1)
PRINT item$(i%), cnt%(i%)/1E6, prob(i%)
NEXT
Output:
Item actual theoretical aleph 0.200306 0.200000 beth 0.165963 0.166667 gimel 0.143089 0.142857 daleth 0.125387 0.125000 he 0.111057 0.111111 waw 0.100098 0.100000 zayin 0.091031 0.090909 heth 0.063069 0.063456
C
#include <stdio.h>
#include <stdlib.h>
/* pick a random index from 0 to n-1, according to probablities listed
in p[] which is assumed to have a sum of 1. The values in the probablity
list matters up to the point where the sum goes over 1 */
int rand_idx(double *p, int n)
{
double s = rand() / (RAND_MAX + 1.0);
int i;
for (i = 0; i < n - 1 && (s -= p[i]) >= 0; i++);
return i;
}
#define LEN 8
#define N 1000000
int main()
{
const char *names[LEN] = { "aleph", "beth", "gimel", "daleth",
"he", "waw", "zayin", "heth" };
double s, p[LEN] = { 1./5, 1./6, 1./7, 1./8, 1./9, 1./10, 1./11, 1e300 };
int i, count[LEN] = {0};
for (i = 0; i < N; i++) count[rand_idx(p, LEN)] ++;
printf(" Name Count Ratio Expected\n");
for (i = 0, s = 1; i < LEN; s -= p[i++])
printf("%6s%7d %7.4f%% %7.4f%%\n",
names[i], count[i],
(double)count[i] / N * 100,
((i < LEN - 1) ? p[i] : s) * 100);
return 0;
}
output
Name Count Ratio Expected
aleph 199928 19.9928% 20.0000%
beth 166489 16.6489% 16.6667%
gimel 143211 14.3211% 14.2857%
daleth 125257 12.5257% 12.5000%
he 110849 11.0849% 11.1111%
waw 99935 9.9935% 10.0000%
zayin 91001 9.1001% 9.0909%
heth 63330 6.3330% 6.3456%
C#
using System;
class Program
{
static long TRIALS = 1000000L;
private class Expv
{
public string name;
public int probcount;
public double expect;
public double mapping;
public Expv(string name, int probcount, double expect, double mapping)
{
this.name = name;
this.probcount = probcount;
this.expect = expect;
this.mapping = mapping;
}
}
static Expv[] items = {
new Expv("aleph", 0, 0.0, 0.0), new Expv("beth", 0, 0.0, 0.0),
new Expv("gimel", 0, 0.0, 0.0), new Expv("daleth", 0, 0.0, 0.0),
new Expv("he", 0, 0.0, 0.0), new Expv("waw", 0, 0.0, 0.0),
new Expv("zayin", 0, 0.0, 0.0), new Expv("heth", 0, 0.0, 0.0)
};
static void Main(string[] args)
{
double rnum, tsum = 0.0;
Random random = new Random();
for (int i = 0, rnum = 5.0; i < 7; i++, rnum += 1.0)
{
items[i].expect = 1.0 / rnum;
tsum += items[i].expect;
}
items[7].expect = 1.0 - tsum;
items[0].mapping = 1.0 / 5.0;
for (int i = 1; i < 7; i++)
items[i].mapping = items[i - 1].mapping + 1.0 / ((double)i + 5.0);
items[7].mapping = 1.0;
for (int i = 0; i < TRIALS; i++)
{
rnum = random.NextDouble();
for (int j = 0; j < 8; j++)
if (rnum < items[j].mapping)
{
items[j].probcount++;
break;
}
}
Console.WriteLine("Trials: {0}", TRIALS);
Console.Write("Items: ");
for (int i = 0; i < 8; i++)
Console.Write(items[i].name.PadRight(9));
Console.WriteLine();
Console.Write("Target prob.: ");
for (int i = 0; i < 8; i++)
Console.Write("{0:0.000000} ", items[i].expect);
Console.WriteLine();
Console.Write("Attained prob.: ");
for (int i = 0; i < 8; i++)
Console.Write("{0:0.000000} ", (double)items[i].probcount / (double)TRIALS);
Console.WriteLine();
}
}
Output:
Trials: 1000000 Items: aleph beth gimel daleth he waw zayin heth Target prob.: 0.200000 0.166667 0.142857 0.125000 0.111111 0.100000 0.090909 0.063456 Attained prob.: 0.199975 0.166460 0.142290 0.125510 0.111374 0.100018 0.090746 0.063627
C++
#include <cstdlib>
#include <iostream>
#include <vector>
#include <utility>
#include <algorithm>
#include <ctime>
#include <iomanip>
int main( ) {
typedef std::vector<std::pair<std::string, double> >::const_iterator SPI ;
typedef std::vector<std::pair<std::string , double> > ProbType ;
ProbType probabilities ;
probabilities.push_back( std::make_pair( "aleph" , 1/5.0 ) ) ;
probabilities.push_back( std::make_pair( "beth" , 1/6.0 ) ) ;
probabilities.push_back( std::make_pair( "gimel" , 1/7.0 ) ) ;
probabilities.push_back( std::make_pair( "daleth" , 1/8.0 ) ) ;
probabilities.push_back( std::make_pair( "he" , 1/9.0 ) ) ;
probabilities.push_back( std::make_pair( "waw" , 1/10.0 ) ) ;
probabilities.push_back( std::make_pair( "zayin" , 1/11.0 ) ) ;
probabilities.push_back( std::make_pair( "heth" , 1759/27720.0 ) ) ;
std::vector<std::string> generated ; //for the strings that are generatod
std::vector<int> decider ; //holds the numbers that determine the choice of letters
for ( int i = 0 ; i < probabilities.size( ) ; i++ ) {
if ( i == 0 ) {
decider.push_back( 27720 * (probabilities[ i ].second) ) ;
}
else {
int number = 0 ;
for ( int j = 0 ; j < i ; j++ ) {
number += 27720 * ( probabilities[ j ].second ) ;
}
number += 27720 * probabilities[ i ].second ;
decider.push_back( number ) ;
}
}
srand( time( 0 ) ) ;
for ( int i = 0 ; i < 1000000 ; i++ ) {
int randnumber = rand( ) % 27721 ;
int j = 0 ;
while ( randnumber > decider[ j ] )
j++ ;
generated.push_back( ( probabilities[ j ]).first ) ;
}
std::cout << "letter frequency attained frequency expected\n" ;
for ( SPI i = probabilities.begin( ) ; i != probabilities.end( ) ; i++ ) {
std::cout << std::left << std::setw( 8 ) << i->first ;
int found = std::count ( generated.begin( ) , generated.end( ) , i->first ) ;
std::cout << std::left << std::setw( 21 ) << found / 1000000.0 ;
std::cout << std::left << std::setw( 17 ) << i->second << '\n' ;
}
return 0 ;
}
Output:
letter frequency attained frequency expected aleph 0.200089 0.2 beth 0.16695 0.166667 gimel 0.142693 0.142857 daleth 0.124859 0.125 he 0.111258 0.111111 waw 0.099665 0.1 zayin 0.090654 0.0909091 heth 0.063832 0.063456
Clojure
Works by first converting the provided Probability Distribution Function into a Cumulative Distribution Function, so that it can simply scan through the CDF list and return the current item as soon as the CDF at that point is greater than the random number generated. The code could be made more concise by skipping this step and instead tracking the whole PDF for each random number; but this code is both faster and more readable.
It uses the language built-in (frequencies) to count the number of occurrences of each distinct name. Note that while we actually generate a sequence of num-trials random samples, the sequence is lazily generated and lazily consumed. This means that the program will scale to an arbitrarily-large num-trials with no ill effects, by throwing away elements it's already processed.
(defn to-cdf [pdf]
(reduce
(fn [acc n] (conj acc (+ (or (last acc) 0) n)))
[]
pdf))
(defn choose [cdf]
(let [r (rand)]
(count
(filter (partial > r) cdf))))
(def *names* '[aleph beth gimel daleth he waw zayin heth])
(def *pdf* (map double [1/5 1/6 1/7 1/8 1/9 1/10 1/11 1759/27720]))
(let [num-trials 1000000
cdf (to-cdf *pdf*)
indexes (range (count *names*)) ;; use integer key internally, not name
expected (into (sorted-map) (zipmap indexes *pdf*))
actual (frequencies (repeatedly num-trials #(choose cdf)))]
(doseq [[idx exp] expected]
(println "Expected number of" (*names* idx) "was"
(* num-trials exp) "and actually got" (actual idx))))
Expected number of aleph was 200000.0 and actually got 199300 Expected number of beth was 166666.66666666672 and actually got 166291 Expected number of gimel was 142857.1428571429 and actually got 143297 Expected number of daleth was 125000.0 and actually got 125032 Expected number of he was 111111.11111111111 and actually got 111540 Expected number of waw was 100000.0 and actually got 100062 Expected number of zayin was 90909.09090909091 and actually got 90719 Expected number of heth was 63455.98845598846 and actually got 63759
Common Lisp
This is a straightforward, if a little verbose implementation based upon the Perl one.
(defvar *probabilities* '((aleph 1/5)
(beth 1/6)
(gimel 1/7)
(daleth 1/8)
(he 1/9)
(waw 1/10)
(zayin 1/11)
(heth 1759/27720)))
(defun calculate-probabilities (choices &key (repetitions 1000000))
(assert (= 1 (reduce #'+ choices :key #'second)))
(labels ((make-ranges ()
(loop for (datum probability) in choices
sum (coerce probability 'double-float) into total
collect (list datum total)))
(pick (ranges)
(declare (optimize (speed 3) (safety 0) (debug 0)))
(loop with random = (random 1.0d0)
for (datum below) of-type (t double-float) in ranges
when (< random below)
do (return datum)))
(populate-hash (ranges)
(declare (optimize (speed 3) (safety 0) (debug 0)))
(loop repeat (the fixnum repetitions)
with hash = (make-hash-table)
do (incf (the fixnum (gethash (pick ranges) hash 0)))
finally (return hash)))
(make-table-data (hash)
(loop for (datum probability) in choices
collect (list datum
(float (/ (gethash datum hash)
repetitions))
(float probability)))))
(format t "Datum~10,2TOccured~20,2TExpected~%")
(format t "~{~{~A~10,2T~F~20,2T~F~}~%~}"
(make-table-data (populate-hash (make-ranges))))))
CL-USER> (calculate-probabilities *probabilities*)
Datum Occured Expected
ALEPH 0.200156 0.2
BETH 0.166521 0.16666667
GIMEL 0.142936 0.14285715
DALETH 0.124779 0.125
HE 0.111601 0.11111111
WAW 0.100068 0.1
ZAYIN 0.090458 0.09090909
HETH 0.063481 0.06345599
D
Basic Version
void main() {
import std.stdio, std.random, std.string, std.range;
enum int nTrials = 1_000_000;
const items = "aleph beth gimel daleth he waw zayin heth".split;
const pr = [1/5., 1/6., 1/7., 1/8., 1/9., 1/10., 1/11., 1759/27720.];
double[pr.length] counts = 0.0;
foreach (immutable _; 0 .. nTrials)
counts[pr.dice]++;
writeln("Item Target prob Attained prob");
foreach (name, p, co; zip(items, pr, counts[]))
writefln("%-7s %.8f %.8f", name, p, co / nTrials);
}
- Output:
Item Target prob Attained prob aleph 0.20000000 0.19964000 beth 0.16666667 0.16753600 gimel 0.14285714 0.14283300 daleth 0.12500000 0.12515400 he 0.11111111 0.11074300 waw 0.10000000 0.10025800 zayin 0.09090909 0.09070400 heth 0.06345598 0.06313200
A Faster Version
void main() {
import std.stdio, std.random, std.algorithm, std.range;
enum int nTrials = 1_000_000;
const items = "aleph beth gimel daleth he waw zayin heth".split;
const pr = [1/5., 1/6., 1/7., 1/8., 1/9., 1/10., 1/11., 1759/27720.];
double[pr.length] cumulatives = pr[];
foreach (immutable i, ref c; cumulatives[1 .. $ - 1])
c += cumulatives[i];
cumulatives[$ - 1] = 1.0;
double[pr.length] counts = 0.0;
auto rnd = Xorshift(unpredictableSeed);
foreach (immutable _; 0 .. nTrials)
counts[cumulatives[].countUntil!(c => c >= rnd.uniform01)]++;
writeln("Item Target prob Attained prob");
foreach (name, p, co; zip(items, pr, counts[]))
writefln("%-7s %.8f %.8f", name, p, co / nTrials);
}
DuckDB
DuckDB does not seem to have a space-efficient way to solve this type of problem. The most noticeable drawback of using DuckDB here, however, appears to be the time it takes to tabulate the results of the random draws. The tabulation is accomplished here using the builtin function histogram().
Because of DuckDB's zealous "inlining", it is currently imperative to "materialize" the random value passed in to the choice() function.
set variable letters = ['aleph', 'beth', 'gimel', 'daleth', 'he', 'waw', 'zayin', 'heth'];
# The list of probabilities - last value must be given but will be ignored:
set variable probs = [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 0];
create or replace function cumulative_probabilities(probs) as (
with recursive cte0 as
(select 1 as ix, probs[1]::double as cum
union all
select ix+1 as ix,
cum + probs[ix + 1] as cum
from cte0
where ix < length(probs))
select array_agg(cum order by ix)
from cte0
);
# It is imperative to materialize rand because of DuckDB's "inlining"
create or replace function choice(rand, cumProbs) as (
with r as materialized(select rand as r)
select
if (r <= cumProbs[1], 1,
if (r <= cumProbs[2], 2,
if (r <= cumProbs[3], 3,
if (r <= cumProbs[4], 4,
if (r <= cumProbs[5], 5,
if (r <= cumProbs[6], 6,
if (r <= cumProbs[7], 7, 8) ))))))
from r
);
# Infer the expected value for the last category
create or replace function expectations(probs, n) as (
with ns as (select list_transform(probs[0:-2], x -> x*n) as ns),
subtotal as (select list_sum(ns) as subtotal from ns)
select ns || [ n - subtotal ]
from ns, subtotal
);
# Generate the histogram of observed letters
create or replace function play(mx) as table (
with recursive cumProbs as (select cumulative_probabilities(getvariable('probs')) as cumProbs),
cte as (
select 1 as n, choice(random(), cumProbs) as cat
from cumProbs
union all
select n+1 as n, choice(random(), cumProbs) as cat
from cte, cumProbs
where n < mx)
select histogram(cat) as histogram
from cte
);
create or replace function synopsis(n) as table (
e as (select expectations(getvariable('probs'),n) as e)
select i as n,
getvariable('letters')[i] as letter,
e[i]::DECIMAL(10,2) as expected,
coalesce(histogram[i],0) as observed,
((observed - expected)^2/expected)::DECIMAL(10,2) as "(o-e)^2/e"
from e, play(n), range(1, 1 + length(getvariable('probs'))) _(i) )
);
from synopsis(1000000);
- Output:
┌───────┬─────────┬───────────────┬──────────┬───────────────┐ │ n │ letter │ expected │ observed │ (o-e)^2/e │ │ int64 │ varchar │ decimal(10,2) │ uint64 │ decimal(10,2) │ ├───────┼─────────┼───────────────┼──────────┼───────────────┤ │ 1 │ aleph │ 200000.00 │ 199670 │ 0.54 │ │ 2 │ beth │ 166666.67 │ 166850 │ 0.20 │ │ 3 │ gimel │ 142857.14 │ 143461 │ 2.55 │ │ 4 │ daleth │ 125000.00 │ 125164 │ 0.22 │ │ 5 │ he │ 111111.11 │ 111297 │ 0.31 │ │ 6 │ waw │ 100000.00 │ 99632 │ 1.35 │ │ 7 │ zayin │ 90909.09 │ 90792 │ 0.15 │ │ 8 │ heth │ 63455.99 │ 63134 │ 1.63 │ └───────┴─────────┴───────────────┴──────────┴───────────────┘
E
This implementation converts the list of probabilities to sub-intervals of [0.0,1.0), then arranges those intervals in a binary tree for searching based on a random number input.
It is rather verbose, due to using the tree rather than a linear search, and having code to print the tree (which was used to debug it).
pragma.syntax("0.9")First, the algorithm:
/** Makes leaves of the binary tree */
def leaf(value) {
return def leaf {
to run(_) { return value }
to __printOn(out) { out.print("=> ", value) }
}
}
/** Makes branches of the binary tree */
def split(leastRight, left, right) {
return def tree {
to run(specimen) {
return if (specimen < leastRight) {
left(specimen)
} else {
right(specimen)
}
}
to __printOn(out) {
out.print(" ")
out.indent().print(left)
out.lnPrint("< ")
out.print(leastRight)
out.indent().lnPrint(right)
}
}
}
def makeIntervalTree(assocs :List[Tuple[any, float64]]) {
def size :int := assocs.size()
if (size > 1) {
def midpoint := size // 2
return split(assocs[midpoint][1], makeIntervalTree(assocs.run(0, midpoint)),
makeIntervalTree(assocs.run(midpoint)))
} else {
def [[value, _]] := assocs
return leaf(value)
}
}
def setupProbabilisticChoice(entropy, table :Map[any, float64]) {
var cumulative := 0.0
var intervalTable := []
for value => probability in table {
intervalTable with= [value, cumulative]
cumulative += probability
}
def total := cumulative
def selector := makeIntervalTree(intervalTable)
return def probChoice {
# Multiplying by the total helps correct for any error in the sum of the inputs
to run() { return selector(entropy.nextDouble() * total) }
to __printOn(out) {
out.print("Probabilistic choice using tree:")
out.indent().lnPrint(selector)
}
}
}Then the test setup:
def rosetta := setupProbabilisticChoice(entropy, def probTable := [
"aleph" => 1/5,
"beth" => 1/6.0,
"gimel" => 1/7.0,
"daleth" => 1/8.0,
"he" => 1/9.0,
"waw" => 1/10.0,
"zayin" => 1/11.0,
"heth" => 0.063455988455988432,
])
var trials := 1000000
var timesFound := [].asMap()
for i in 1..trials {
if (i % 1000 == 0) { print(`${i//1000} `) }
def value := rosetta()
timesFound with= (value, timesFound.fetch(value, fn { 0 }) + 1)
}
stdout.println()
for item in probTable.domain() {
stdout.print(item, "\t", timesFound[item] / trials, "\t", probTable[item], "\n")
}EasyLang
name$[] = [ "aleph " "beth " "gimel " "daleth" "he " "waw " "zayin " "heth " ]
probs[] = [ 1 / 5 1 / 6 1 / 7 1 / 8 1 / 9 1 / 10 1 / 11 0 ]
for i = 1 to 7
cum += probs[i]
cum[] &= cum
.
cum[] &= 1
probs[8] = 1 - cum[7]
len act[] 8
n = 1000000
for i to n
h = randomf
j = 1
while h > cum[j]
j += 1
.
act[j] += 1
.
print "Name Ratio Expected"
print "---------------------"
numfmt 4 6
for i to 8
print name$[i] & " " & act[i] / n & " " & probs[i]
.
- Output:
Name Ratio Expected --------------------- aleph 0.2000 0.2000 beth 0.1661 0.1667 gimel 0.1424 0.1429 daleth 0.1252 0.1250 he 0.1111 0.1111 waw 0.1003 0.1000 zayin 0.0913 0.0909 heth 0.0637 0.0635
Elixir
defmodule Probabilistic do
@tries 1000000
@probs [aleph: 1/5,
beth: 1/6,
gimel: 1/7,
daleth: 1/8,
he: 1/9,
waw: 1/10,
zayin: 1/11,
heth: 1759/27720]
def test do
trials = for _ <- 1..@tries, do: get_choice(@probs, :rand.uniform)
IO.puts "Item Expected Actual"
fmt = " ~-8s ~.6f ~.6f~n"
Enum.each(@probs, fn {glyph,expected} ->
actual = length(for ^glyph <- trials, do: glyph) / @tries
:io.format fmt, [glyph, expected, actual]
end)
end
defp get_choice([{glyph,_}], _), do: glyph
defp get_choice([{glyph,prob}|_], ran) when ran < prob, do: glyph
defp get_choice([{_,prob}|t], ran), do: get_choice(t, ran - prob)
end
Probabilistic.test
- Output:
Item Expected Actual aleph 0.200000 0.200676 beth 0.166667 0.166103 gimel 0.142857 0.142543 daleth 0.125000 0.125055 he 0.111111 0.111165 waw 0.100000 0.100439 zayin 0.090909 0.090894 heth 0.063456 0.063125
Erlang
The optimized version of Java.
-module(probabilistic_choice).
-export([test/0]).
-define(TRIES, 1000000).
test() ->
Probs =
[{aleph,1/5},
{beth,1/6},
{gimel,1/7},
{daleth,1/8},
{he,1/9},
{waw,1/10},
{zayin,1/11},
{heth,1759/27720}],
random:seed(now()),
Trials =
[get_choice(Probs,random:uniform()) || _ <- lists:seq(1,?TRIES)],
[{Glyph,Expected,(length([Glyph || Glyph_ <- Trials, Glyph_ == Glyph])/?TRIES)}
|| {Glyph,Expected} <- Probs].
get_choice([{Glyph,_}],_) ->
Glyph;
get_choice([{Glyph,Prob}|T],Ran) ->
case (Ran < Prob) of
true ->
Glyph;
false ->
get_choice(T,Ran - Prob)
end.
Output:
[{aleph,0.2,0.200325},
{beth,0.16666666666666666,0.167108},
{gimel,0.14285714285714285,0.142246},
{daleth,0.125,0.124851},
{he,0.1111111111111111,0.111345},
{waw,0.1,0.099912},
{zayin,0.09090909090909091,0.091352},
{heth,0.06345598845598846,0.062861}]
ERRE
PROGRAM PROB_CHOICE
DIM ITEM$[7],PROB[7],CNT[7]
BEGIN
ITEM$[]=("aleph","beth","gimel","daleth","he","waw","zayin","heth")
PROB[0]=1/5.0 PROB[1]=1/6.0 PROB[2]=1/7.0 PROB[3]=1/8.0
PROB[4]=1/9.0 PROB[5]=1/10.0 PROB[6]=1/11.0 PROB[7]=1759/27720
SUM=0
FOR I%=0 TO UBOUND(PROB,1) DO
SUM=SUM+PROB[I%]
END FOR
IF ABS(SUM-1)>1E-6 THEN
PRINT("Probabilities don't sum to 1")
ELSE
FOR TRIAL=1 TO 1E6 DO
R=RND(1)
P=0
FOR I%=0 TO UBOUND(PROB,1) DO
P+=PROB[I%]
IF R<P THEN
CNT[I%]+=1
EXIT
END IF
END FOR
END FOR
PRINT("Item actual theoretical")
PRINT("---------------------------------")
FOR I%=0 TO UBOUND(ITEM$,1) DO
WRITE("\ \ #.###### #.######";ITEM$[I%],CNT[I%]/1E6,PROB[I%])
END FOR
END IF
END PROGRAMOutput:
Item actual theoretical --------------------------------- aleph 0.199769 0.200000 beth 0.167277 0.166667 gimel 0.142914 0.142857 daleth 0.124991 0.125000 he 0.111227 0.111111 waw 0.099732 0.100000 zayin 0.090757 0.090909 heth 0.063333 0.063456
Euphoria
constant MAX = #3FFFFFFF
constant times = 1e6
atom d,e
sequence Mapps
Mapps = {
{ "aleph", 1/5, 0},
{ "beth", 1/6, 0},
{ "gimel", 1/7, 0},
{ "daleth", 1/8, 0},
{ "he", 1/9, 0},
{ "waw", 1/10, 0},
{ "zayin", 1/11, 0},
{ "heth", 1759/27720, 0}
}
for i = 1 to times do
d = (rand(MAX)-1)/MAX
e = 0
for j = 1 to length(Mapps) do
e += Mapps[j][2]
if d <= e then
Mapps[j][3] += 1
exit
end if
end for
end for
printf(1,"Sample times: %d\n",times)
for j = 1 to length(Mapps) do
d = Mapps[j][3]/times
printf(1,"%-7s should be %f is %f | Deviatation %6.3f%%\n",
{Mapps[j][1],Mapps[j][2],d,(1-Mapps[j][2]/d)*100})
end forOutput:
Sample times: 1000000 aleph should be 0.200000 is 0.200492 | Deviatation 0.245% beth should be 0.166667 is 0.166855 | Deviatation 0.113% gimel should be 0.142857 is 0.143169 | Deviatation 0.218% daleth should be 0.125000 is 0.124923 | Deviatation -0.062% he should be 0.111111 is 0.110511 | Deviatation -0.543% waw should be 0.100000 is 0.099963 | Deviatation -0.037% zayin should be 0.090909 is 0.090647 | Deviatation -0.289% heth should be 0.063456 is 0.063440 | Deviatation -0.025%
F#
// Probabilistic choice. Nigel Galloway: November 15th., 2024
type items=Aleph|Beth|Gimel|Daleth|He|Waw|Zayin|Heth
let item=function n when n<5544 ->Aleph
|n when n<10164->Beth
|n when n<14124->Gimel
|n when n<17589->Daleth
|n when n<20669->He
|n when n<23441->Waw
|n when n<25961->Zayin
|_->Heth
let R=System.Random()
let expected=function Aleph ->1000000/5
|Beth ->1000000/6
|Gimel ->1000000/7
|Daleth->1000000/8
|He ->1000000/9
|Waw ->1000000/10
|Zayin ->1000000/11
|Heth ->(1000000*1759)/27720
Seq.init 1000000 (fun _->R.Next(0,27720))|>Seq.countBy item|>Seq.iter(fun(n,g)->printfn $"item={n} count={g} expected={expected n}")
- Output:
item=Beth count=166817 expected=166666 item=Aleph count=199716 expected=200000 item=Zayin count=91278 expected=90909 item=He count=111620 expected=111111 item=Gimel count=142710 expected=142857 item=Daleth count=124716 expected=125000 item=Heth count=63376 expected=63455 item=Waw count=99767 expected=100000
Factor
USING: arrays assocs combinators.random io kernel macros math
math.statistics prettyprint quotations sequences sorting formatting ;
IN: rosettacode.proba
CONSTANT: data
{
{ "aleph" 1/5.0 }
{ "beth" 1/6.0 }
{ "gimel" 1/7.0 }
{ "daleth" 1/8.0 }
{ "he" 1/9.0 }
{ "waw" 1/10.0 }
{ "zayin" 1/11.0 }
{ "heth" f }
}
MACRO: case-probas ( data -- case-probas )
[ first2 [ swap 1quotation 2array ] [ 1quotation ] if* ] map 1quotation ;
: expected ( name data -- float )
2dup at [ 2nip ] [ nip values sift sum 1 swap - ] if* ;
: generate ( # case-probas -- seq )
H{ } clone
[ [ [ casep ] [ inc-at ] bi* ] 2curry times ] keep ; inline
: normalize ( seq # -- seq )
[ clone ] dip [ /f ] curry assoc-map ;
: summarize1 ( name value data -- )
[ over ] dip expected
"%6s: %10f %10f\n" printf ;
: summarize ( generated data -- )
"Key" "Value" "expected" "%6s %10s %10s\n" printf
[ summarize1 ] curry assoc-each ;
: generate-normalized ( # proba -- seq )
[ generate ] [ drop normalize ] 2bi ; inline
: example ( # data -- )
[ case-probas generate-normalized ]
[ summarize ] bi ; inline
In a REPL:
USE: rosettacode.proba
1000000 data example
outputs
Key Value expected
heth: 0.063469 0.063456
waw: 0.100226 0.100000
daleth: 0.125844 0.125000
beth: 0.166264 0.166667
zayin: 0.090806 0.090909
he: 0.110562 0.111111
aleph: 0.199868 0.200000
gimel: 0.142961 0.142857
Fermat
trials:=1000000;
Array probs[8]; {store the probabilities}
[probs]:=[<i=1,8> 1/(i+4)];
probs[8]:=1-Sigma<i=1,7>[probs[i,1]];
Func Round( a, b ) = (2*a+b)\(2*b).; {rounds a fraction with numerator a and denominator b}
; {to the nearest integer (positive fractions only)}
Func Sel = {select a number from 1 to 8 according to the}
r:=Rand|27720; {specified probabilities}
if r < probs[1]*27720 then Return(1) fi;
if r < Sigma<i=1,2>[probs[i]]*27720 then Return(2) fi;
if r < Sigma<i=1,3>[probs[i]]*27720 then Return(3) fi;
if r < Sigma<i=1,4>[probs[i]]*27720 then Return(4) fi;
if r < Sigma<i=1,5>[probs[i]]*27720 then Return(5) fi;
if r < Sigma<i=1,6>[probs[i]]*27720 then Return(6) fi;
if r < Sigma<i=1,7>[probs[i]]*27720 then Return(7) fi;
Return(8);
.;
Array label[10]; {strings are not Fermat's strong suit}
Func Letter(n) = {assign a Hebrew letter to the numbers 1-8}
[label]:='heth ';
if n = 1 then [label]:='aleph ' fi;
if n = 2 then [label]:='beth ' fi;
if n = 3 then [label]:='gimel ' fi;
if n = 4 then [label]:='daleth ' fi;
if n = 5 then [label]:='he ' fi;
if n = 6 then [label]:='waw ' fi;
if n = 7 then [label]:='zayin ' fi;
.;
Array count[8]; {pick a bunch of random numbers}
for i = 1 to trials do
s:=Sel;
count[s]:=count[s]+1;
od;
for i = 1 to 8 do {now display some diagnostics}
Letter(i);
ctp:=count[i]/trials-probs[i];
!([label:char, count[i]/trials,' differs from ',probs[i]);
!(' by ',ctp, ' or about one part in ', Round(Denom(ctp),|Numer(ctp)|));
!!;
od;
!!('The various probabilities add up to ',Sigma<i=1,8>[count[i]/trials]); {check if our trials add to 1}- Output:
aleph 199939 / 1000000 differs from 1 / 5 by -61 / 1000000 or about one part in 16393
beth 166361 / 1000000 differs from 1 / 6 by -917 / 3000000 or about one part in 3272
gimel 142509 / 1000000 differs from 1 / 7 by -2437 / 7000000 or about one part in 2872
daleth 124917 / 1000000 differs from 1 / 8 by -83 / 1000000 or about one part in 12048
he 111013 / 1000000 differs from 1 / 9 by -883 / 9000000 or about one part in 10193
waw 100327 / 1000000 differs from 1 / 10 by 327 / 1000000 or about one part in 3058
zayin 4569 / 50000 differs from 1 / 11 by 259 / 550000 or about one part in 2124
heth 31777 / 500000 differs from 1759 / 27720 by 33961 / 346500000 or about one part in 10203
The various probabilities add up to 1
Forth
include random.fs
\ common factors of desired probabilities (1/5 .. 1/11)
2 2 * 2 * 3 * 3 * 5 * 7 * 11 * constant denom \ 27720
\ represent each probability as the numerator with 27720 as the denominator
: ,numerators ( max min -- )
do denom i / , loop ;
\ final item is 27720 - sum(probs)
: ,remainder ( denom addr len -- )
cells bounds do i @ - 1 cells +loop , ;
create probs 12 5 ,numerators denom probs 7 ,remainder
create bins 8 cells allot
: choose ( -- 0..7 )
denom random
8 0 do
probs i cells + @ -
dup 0< if drop i unloop exit then
loop
abort" can't get here" ;
: trials ( n -- )
0 do 1 bins choose cells + +! loop ;
: str-table
create ( c-str ... n -- ) 0 do , loop
does> ( n -- str len ) swap cells + @ count ;
here ," heth" here ," zayin" here ," waw" here ," he"
here ," daleth" here ," gimel" here ," beth" here ," aleph"
8 str-table names
: .header
cr ." Name" #tab emit ." Prob" #tab emit ." Actual" #tab emit ." Error" ;
: .result ( n -- )
cr dup names type #tab emit
dup cells probs + @ s>f denom s>f f/ fdup f. #tab emit
dup cells bins + @ s>f 1e6 f/ fdup f. #tab emit
f- fabs fs. ;
: .results .header 8 0 do i .result loop ;
bins 8 cells erase 3 set-precision 1000000 trials .results Name Prob Actual Error aleph 0.2 0.2 9.90E-5 beth 0.167 0.167 4.51E-4 gimel 0.143 0.142 4.99E-4 daleth 0.125 0.125 1.82E-4 he 0.111 0.111 2.10E-4 waw 0.1 0.1 3.30E-5 zayin 0.0909 0.0912 2.77E-4 heth 0.0635 0.0636 9.70E-5 ok
Fortran
PROGRAM PROBS
IMPLICIT NONE
INTEGER, PARAMETER :: trials = 1000000
INTEGER :: i, j, probcount(8) = 0
REAL :: expected(8), mapping(8), rnum
CHARACTER(6) :: items(8) = (/ "aleph ", "beth ", "gimel ", "daleth", "he ", "waw ", "zayin ", "heth " /)
expected(1:7) = (/ (1.0/i, i=5,11) /)
expected(8) = 1.0 - SUM(expected(1:7))
mapping(1) = 1.0 / 5.0
DO i = 2, 7
mapping(i) = mapping(i-1) + 1.0/(i+4.0)
END DO
mapping(8) = 1.0
DO i = 1, trials
CALL RANDOM_NUMBER(rnum)
DO j = 1, 8
IF (rnum < mapping(j)) THEN
probcount(j) = probcount(j) + 1
EXIT
END IF
END DO
END DO
WRITE(*, "(A,I10)") "Trials: ", trials
WRITE(*, "(A,8A10)") "Items: ", items
WRITE(*, "(A,8F10.6)") "Target Probability: ", expected
WRITE(*, "(A,8F10.6)") "Attained Probability:", REAL(probcount) / REAL(trials)
ENDPROGRAM PROBS
Sample Output:
Trials: 1000000 Items: aleph beth gimel daleth he waw zayin heth Target Probability: 0.200000 0.166667 0.142857 0.125000 0.111111 0.100000 0.090909 0.063456 Attained Probability: 0.199631 0.166907 0.142488 0.124920 0.110906 0.099943 0.091775 0.063430
FreeBASIC
' FB 1.05.0 Win64
Dim letters (0 To 7) As String = {"aleph", "beth", "gimel", "daleth", "he", "waw", "zayin", "heth"}
Dim actual (0 To 7) As Integer '' all zero by default
Dim probs (0 To 7) As Double = {1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0}
Dim cumProbs (0 To 7) As Double
cumProbs(0) = probs(0)
For i As Integer = 1 To 6
cumProbs(i) = cumProbs(i - 1) + probs(i)
Next
cumProbs(7) = 1.0
probs(7) = 1.0 - cumProbs(6)
Randomize
Dim rand As Double
Dim n As Double = 1000000
Dim sum As Double = 0.0
For i As Integer = 1 To n
rand = Rnd '' random number where 0 <= rand < 1
Select case rand
Case Is <= cumProbs(0)
actual(0) += 1
Case Is <= cumProbs(1)
actual(1) += 1
Case Is <= cumProbs(2)
actual(2) += 1
Case Is <= cumProbs(3)
actual(3) += 1
Case Is <= cumProbs(4)
actual(4) += 1
Case Is <= cumProbs(5)
actual(5) += 1
Case Is <= cumProbs(6)
actual(6) += 1
Case Else
actual(7) += 1
End Select
Next
Dim sumActual As Double = 0
Print "Letter", " Actual", "Expected"
Print "------", "--------", "--------"
For i As Integer = 0 To 7
Print letters(i),
Print Using "#.######"; actual(i)/n;
sumActual += actual(i)/n
Print , Using "#.######"; probs(i)
Next
Print , "--------", "--------"
Print , Using "#.######"; sumActual;
Print , Using "#.######"; 1.000000
Print
Print "Press any key to quit"
Sleep
- Output:
Letter Actual Expected
------ -------- --------
aleph 0.199987 0.200000
beth 0.166663 0.166667
gimel 0.143134 0.142857
daleth 0.125132 0.125000
he 0.110772 0.111111
waw 0.100236 0.100000
> 0.090664 0.090909
heth 0.063412 0.063456
-------- --------
1.000000 1.000000
FutureBasic
_elements = 8
local fn ProbabilisticChoice
double prob(_elements), cumulative(_elements)
Str15 item(_elements)
double r, p, sum = 0, checksum = 0
long i, j, samples = 1000000
item(1) = "aleph" : item(2) = "beth" : item(3) = "gimel" : item(4) = "daleth"
item(5) = "he" : item(6) = "waw" : item(7) = "zayin" : item(8) = "heth"
prob(1) = 1/5.0 : prob(2) = 1/6.0 : prob(3) = 1/7.0 : prob(4) = 1/8.0
prob(5) = 1/9.0 : prob(6) = 1/10.0 : prob(7) = 1/11.0 : prob(8) = 1759/27720
for i = 1 to _elements
sum += prob(i)
next
if abs(sum-1) > samples then print "Probabilities don't sum to 1." : exit fn
for i = 1 to samples
cln r = (((double)arc4random()/0x100000000));
p = 0
for j = 1 to _elements
p += prob(j)
if (r < p) then cumulative(j) += 1 : exit for
next
next
print
printf @"Item Actual Theoretical"
printf @"---- ------ -----------"
for i = 1 to _elements
printf @"%-7s %10.6f %12.6f", item(i), cumulative(i)/samples, prob(i)
checksum += cumulative(i)/samples
next
printf @" -------- -----------"
printf @"%17.6f %12.6f", checksum, 1.000000
end fn
fn ProbabilisticChoice
HandleEvents- Output:
Item Actual Theoretical
---- ------ -----------
aleph 0.199966 0.200000
beth 0.166505 0.166667
gimel 0.142958 0.142857
daleth 0.124827 0.125000
he 0.111451 0.111111
waw 0.100095 0.100000
zayin 0.090985 0.090909
heth 0.063213 0.063456
-------- -----------
1.000000 1.000000
Go
package main
import (
"fmt"
"math/rand"
"time"
)
type mapping struct {
item string
pr float64
}
func main() {
// input mapping
m := []mapping{
{"aleph", 1 / 5.},
{"beth", 1 / 6.},
{"gimel", 1 / 7.},
{"daleth", 1 / 8.},
{"he", 1 / 9.},
{"waw", 1 / 10.},
{"zayin", 1 / 11.},
{"heth", 1759 / 27720.}} // adjusted so that probabilities add to 1
// cumulative probability
cpr := make([]float64, len(m)-1)
var c float64
for i := 0; i < len(m)-1; i++ {
c += m[i].pr
cpr[i] = c
}
// generate
const samples = 1e6
occ := make([]int, len(m))
rand.Seed(time.Now().UnixNano())
for i := 0; i < samples; i++ {
r := rand.Float64()
for j := 0; ; j++ {
if r < cpr[j] {
occ[j]++
break
}
if j == len(cpr)-1 {
occ[len(cpr)]++
break
}
}
}
// report
fmt.Println(" Item Target Generated")
var totalTarget, totalGenerated float64
for i := 0; i < len(m); i++ {
target := m[i].pr
generated := float64(occ[i]) / samples
fmt.Printf("%6s %8.6f %8.6f\n", m[i].item, target, generated)
totalTarget += target
totalGenerated += generated
}
fmt.Printf("Totals %8.6f %8.6f\n", totalTarget, totalGenerated)
}
Output:
Item Target Generated
aleph 0.200000 0.199509
beth 0.166667 0.167194
gimel 0.142857 0.143293
daleth 0.125000 0.124869
he 0.111111 0.110896
waw 0.100000 0.099849
zayin 0.090909 0.090789
heth 0.063456 0.063601
Totals 1.000000 1.000000
Haskell
import System.Random (newStdGen, randomRs)
dataBinCounts :: [Float] -> [Float] -> [Int]
dataBinCounts thresholds range =
zipWith
(-)
(xs <> [sampleSize])
(0 : xs)
where
sampleSize = length range
xs =
(-) sampleSize . length
. flip filter range
. (<)
<$> thresholds
--------------------------- TEST -------------------------
main :: IO ()
main = do
g <- newStdGen
let fractions = recip <$> [5 .. 11] :: [Float]
expected = fractions <> [1 - sum fractions]
actual =
(/ 1000000.0) . fromIntegral
<$> dataBinCounts
(scanl1 (+) expected)
(take 1000000 (randomRs (0, 1) g))
piv n x = take n (x <> repeat ' ');
putStrLn " expected actual"
mapM_ putStrLn $
zipWith3
( \l s c ->
piv 7 l
<> (piv 13 (show s) <> piv 12 (show c))
)
[ "aleph",
"beth",
"gimel",
"daleth",
"he",
"waw",
"zayin",
"heth"
]
expected
actual
- Output:
Sample
expected actual aleph 0.2 0.200597 beth 0.16666667 0.167192 gimel 0.14285715 0.142781 daleth 0.125 0.124556 he 0.11111111 0.111128 waw 0.1 9.9671e-2 zayin 9.090909e-2 9.0294e-2 heth 6.345594e-2 6.3781e-2
HicEst
REAL :: trials=1E6, n=8, map(n), limit(n), expected(n), outcome(n)
expected = 1 / ($ + 4)
expected(n) = 1 - SUM(expected) + expected(n)
map = expected
map = map($) + map($-1)
DO i = 1, trials
random = RAN(1)
limit = random > map
item = INDEX(limit, 0)
outcome(item) = outcome(item) + 1
ENDDO
outcome = outcome / trials
DLG(Text=expected, Text=outcome, Y=0)Exported from the spreadsheet-like DLG function:
0.2 0.199908 0.1666667 0.166169 0.1428571 0.142722 0.125 0.124929 0.1111111 0.111706 0.1 0.099863 0.0909091 0.090965 0.063456 0.063738
Icon and Unicon
Output:
aleph 0.2 0.1988
beth 0.1666666667 0.1676
gimel 0.1428571429 0.1431
daleth 0.125 0.1249
he 0.1111111111 0.1112
waw 0.1 0.0996
zayin 0.09090909091 0.0908
heth 0.06345598846 0.0636
J
main=: verb define
hdr=. ' target actual '
lbls=. ; ,:&.> ;:'aleph beth gimel daleth he waw zayin heth'
prtn=. +/\ pt=. (, 1-+/)1r1%5+i.7
da=. prtn I. ?y # 0
pa=. y%~ +/ da =/ i.8
hdr, lbls,. 9j6 ": |: pt,:pa
)
Note 'named abbreviations'
hdr (header)
lbls (labels)
pt (target proportions)
prtn (partitions corresponding to target proportions)
da (distribution of actual values among partitions)
pa (actual proportions)
)
Example use:
main 1e6
target actual
aleph 0.200000 0.200344
beth 0.166667 0.166733
gimel 0.142857 0.142611
daleth 0.125000 0.124458
he 0.111111 0.111455
waw 0.100000 0.099751
zayin 0.090909 0.091121
heth 0.063456 0.063527
Note that there is no rounding error in summing the proportions, as they are represented as rational numbers, not floating-point approximations.
pt=. (, 1-+/)1r1%5+i.7
pt
1r5 1r6 1r7 1r8 1r9 1r10 1r11 1759r27720
+/pt
1
Java
public class Prob{
static long TRIALS= 1000000;
private static class Expv{
public String name;
public int probcount;
public double expect;
public double mapping;
public Expv(String name, int probcount, double expect, double mapping){
this.name= name;
this.probcount= probcount;
this.expect= expect;
this.mapping= mapping;
}
}
static Expv[] items=
{new Expv("aleph", 0, 0.0, 0.0), new Expv("beth", 0, 0.0, 0.0),
new Expv("gimel", 0, 0.0, 0.0),
new Expv("daleth", 0, 0.0, 0.0),
new Expv("he", 0, 0.0, 0.0), new Expv("waw", 0, 0.0, 0.0),
new Expv("zayin", 0, 0.0, 0.0),
new Expv("heth", 0, 0.0, 0.0)};
public static void main(String[] args){
int i, j;
double rnum, tsum= 0.0;
for(i= 0, rnum= 5.0;i < 7;i++, rnum+= 1.0){
items[i].expect= 1.0 / rnum;
tsum+= items[i].expect;
}
items[7].expect= 1.0 - tsum;
items[0].mapping= 1.0 / 5.0;
for(i= 1;i < 7;i++){
items[i].mapping= items[i - 1].mapping + 1.0 / ((double)i + 5.0);
}
items[7].mapping= 1.0;
for(i= 0;i < TRIALS;i++){
rnum= Math.random();
for(j= 0;j < 8;j++){
if(rnum < items[j].mapping){
items[j].probcount++;
break;
}
}
}
System.out.printf("Trials: %d\n", TRIALS);
System.out.printf("Items: ");
for(i= 0;i < 8;i++)
System.out.printf("%-8s ", items[i].name);
System.out.printf("\nTarget prob.: ");
for(i= 0;i < 8;i++)
System.out.printf("%8.6f ", items[i].expect);
System.out.printf("\nAttained prob.: ");
for(i= 0;i < 8;i++)
System.out.printf("%8.6f ", (double)(items[i].probcount)
/ (double)TRIALS);
System.out.printf("\n");
}
}
Output:
Trials: 1000000 Items: aleph beth gimel daleth he waw zayin heth Target prob.: 0.200000 0.166667 0.142857 0.125000 0.111111 0.100000 0.090909 0.063456 Attained prob.: 0.199615 0.167517 0.142612 0.125211 0.110970 0.099614 0.091002 0.063459
import java.util.EnumMap;
public class Prob {
public static long TRIALS= 1000000;
public enum Glyph{
ALEPH, BETH, GIMEL, DALETH, HE, WAW, ZAYIN, HETH;
}
public static EnumMap<Glyph, Double> probs = new EnumMap<Glyph, Double>(Glyph.class){{
put(Glyph.ALEPH, 1/5.0);
put(Glyph.BETH, 1/6.0);
put(Glyph.GIMEL, 1/7.0);
put(Glyph.DALETH, 1/8.0);
put(Glyph.HE, 1/9.0);
put(Glyph.WAW, 1/10.0);
put(Glyph.ZAYIN, 1/11.0);
put(Glyph.HETH, 1759./27720);
}};
public static EnumMap<Glyph, Double> counts = new EnumMap<Glyph, Double>(Glyph.class){{
put(Glyph.ALEPH, 0.);put(Glyph.BETH, 0.);
put(Glyph.GIMEL, 0.);put(Glyph.DALETH, 0.);
put(Glyph.HE, 0.);put(Glyph.WAW, 0.);
put(Glyph.ZAYIN, 0.);put(Glyph.HETH, 0.);
}};
public static void main(String[] args){
System.out.println("Target probabliities:\t" + probs);
for(long i = 0; i < TRIALS; i++){
Glyph choice = getChoice();
counts.put(choice, counts.get(choice) + 1);
}
//correct the counts to probablities in (0..1]
for(Glyph glyph:counts.keySet()){
counts.put(glyph, counts.get(glyph) / TRIALS);
}
System.out.println("Actual probabliities:\t" + counts);
}
private static Glyph getChoice() {
double rand = Math.random();
for(Glyph item:Glyph.values()){
if(rand < probs.get(item)){
return item;
}
rand -= probs.get(item);
}
return null;
}
}
Output:
Target probabliities: {ALEPH=0.2, BETH=0.16666666666666666, GIMEL=0.14285714285714285, DALETH=0.125, HE=0.1111111111111111, WAW=0.1, ZAYIN=0.09090909090909091, HETH=0.06345598845598846}
Actual probabliities: {ALEPH=0.200794, BETH=0.165916, GIMEL=0.143286, DALETH=0.124727, HE=0.110818, WAW=0.100168, ZAYIN=0.090878, HETH=0.063413}
JavaScript
ES5
Fortunately, iterating over properties added to an object maintains the insertion order.
var probabilities = {
aleph: 1/5.0,
beth: 1/6.0,
gimel: 1/7.0,
daleth: 1/8.0,
he: 1/9.0,
waw: 1/10.0,
zayin: 1/11.0,
heth: 1759/27720
};
var sum = 0;
var iterations = 1000000;
var cumulative = {};
var randomly = {};
for (var name in probabilities) {
sum += probabilities[name];
cumulative[name] = sum;
randomly[name] = 0;
}
for (var i = 0; i < iterations; i++) {
var r = Math.random();
for (var name in cumulative) {
if (r <= cumulative[name]) {
randomly[name]++;
break;
}
}
}
for (var name in probabilities)
// using WSH
WScript.Echo(name + "\t" + probabilities[name] + "\t" + randomly[name]/iterations);
output:
aleph 0.2 0.200597 beth 0.16666666666666666 0.166527 gimel 0.14285714285714285 0.142646 daleth 0.125 0.124613 he 0.1111111111111111 0.111342 waw 0.1 0.099888 zayin 0.09090909090909091 0.091141 heth 0.06345598845598846 0.063246
ES6
By functional composition:
(() => {
'use strict';
// GENERIC FUNCTIONS -----------------------------------------------------
// transpose :: [[a]] -> [[a]]
const transpose = xs =>
xs[0].map((_, iCol) => xs.map(row => row[iCol]));
// justifyLeft :: Int -> Char -> Text -> Text
const justifyLeft = (n, cFiller, strText) =>
n > strText.length ? (
(strText + cFiller.repeat(n))
.substr(0, n)
) : strText;
// 2 or more arguments
// curry :: Function -> Function
const curry = (f, ...args) => {
const go = xs => xs.length >= f.length ? (f.apply(null, xs)) :
function () {
return go(xs.concat([].slice.apply(arguments)));
};
return go([].slice.call(args, 1));
};
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = (f, xs, ys) => {
const ny = ys.length;
return (xs.length <= ny ? xs : xs.slice(0, ny))
.map((x, i) => f(x, ys[i]));
};
// subtract :: (Num a) => a -> a -> a
const subtract = (x, y) => y - x;
// scanl1 :: (a -> a -> a) -> [a] -> [a]
const scanl1 = (f, xs) =>
xs.length > 0 ? scanl(f, xs[0], xs.slice(1)) : [];
// scanl :: (b -> a -> b) -> b -> [a] -> [b]
const scanl = (f, startValue, xs) =>
xs.reduce((a, x) => {
const v = f(a.acc, x);
return {
acc: v,
scan: a.scan.concat(v)
};
}, {
acc: startValue,
scan: [startValue]
})
.scan;
// unwords :: [String] -> String
const unwords = xs => xs.join(' ');
// PROBABILISTIC CHOICE --------------------------------------------------
// samples :: Int -> Int -> [Float]
const samples = n =>
Array.from({
length: n
}, Math.random);
// thresholds :: Float
const thresholds = scanl1(
(a, b) => a + b, [5, 6, 7, 8, 9, 10, 11].map(x => 1 / x)
)
.concat(1);
// expected :: Float -> Float
const expected = limits =>
limits.map((x, i, xs) => i > 0 ? (x - xs[i - 1]) : x);
// dataBinCounts :: [Float] -> [Float] -> [Int]
const dataBinCounts = (thresholds, samples) => {
const
lng = samples.length,
xs = thresholds
.map(x => lng - samples.filter(v => v > x)
.length);
return zipWith(subtract, [0].concat(xs), xs.concat(lng));
};
// intSamples :: Integer
const intSamples = 1000000;
// aligned :: a -> String
const aligned = x => justifyLeft(12, ' ', isNaN(x) ? x : x.toFixed(7));
return transpose([
['', 'Aleph', 'Beit', 'Gimel', 'Dalet', 'He', 'Vav', 'Zayin', 'Chet']
.map(curry(justifyLeft)(7, ' ')),
['Expected'].concat(expected(thresholds))
.map(aligned),
['Observed'].concat(dataBinCounts(thresholds, samples(intSamples))
.map(x => x / intSamples))
.map(aligned)
])
.map(unwords)
.join('\n');
})();
- Output:
Sample:
Expected Observed Aleph 0.2000000 0.2002440 Beit 0.1666667 0.1665330 Gimel 0.1428571 0.1433880 Dalet 0.1250000 0.1244630 He 0.1111111 0.1112830 Vav 0.1000000 0.0998390 Zayin 0.0909091 0.0909630 Chet 0.0634560 0.0632870
jq
In the task description, the probabilities are given as rationals, so in this entry, we shall assume the probabilities are given as rationals represented by JSON objects of the form {n: $numerator, d: denominator}. For simplicity, it is further assumed that the largest denominator is not too large.
Neither the C nor the Go implmentations of jq provides a PRNG, so in the following, /dev/urandom is used as a source of entropy; an appropriate invocation of jq (or gojq) would thus be as follows:
#!/bin/bash
< /dev/urandom tr -cd '0-9' | fold -w 1 | jq -nr -f probabilistic-choice.jq
# Output: a prn in range(0;$n) where $n is `.`
def prn:
if . == 1 then 0
else . as $n
| ([1, (($n-1)|tostring|length)]|max) as $w
| [limit($w; inputs)] | join("") | tonumber
| if . < $n then . else ($n | prn) end
end;
# General Utility Functions
# bag of words
def bow(stream):
reduce stream as $word ({}; .[($word|tostring)] += 1);
# left pad with blank
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
# right-pad with 0
def rpad($len): tostring | ($len - length) as $l | .+ ("0" * $l)[:$l];
# Input: a string of digits with up to one "."
# Output: the corresponding string representation with exactly $n decimal digits
def align_decimal($n):
tostring
| index(".") as $ix
| if $ix then capture("(?<i>[0-9]*[.])(?<j>[0-9]{0," + ($n|tostring) + "})") as {$i, $j}
| $i + ($j|rpad($n))
else . + "." + ("0" * $n)
end ;
# Input: a string of digits with up to one embedded "."
# Output: the corresponding string representation with up to $n decimal digits but aligned at the period
def align_decimal($n):
tostring
| index(".") as $ix
| if $ix then capture("(?<i>[0-9]*[.])(?<j>[0-9]{0," + ($n|tostring) + "})") as {$i, $j}
| $i + ($j|rpad($n))
else . + ".0" | align_decimal($n)
end ;
# least common multiple
# Define the helper function to take advantage of jq tail-recursion optimization
def lcm($m; $n):
def _lcm:
# state is [m, n, i]
if (.[2] % .[1]) == 0 then .[2]
else .[0:2] + [.[2] + $m] | _lcm
end;
[m, n, m] | _lcm;
def lcm(s): reduce s as $_ (1; lcm(.; $_));
# rationals
def r($n; $d): {$n, $d};The Task
# Given that $integers is an array of integers to be interpreted as
# relative probabilities, return a corresponding element chosen
# randomly from the input array.
def randomly($integers):
def accumulate: reduce .[1:][] as $i ([.[0]]; . + [$i + .[-1]]);
if ($integers | length) != length
then "randomly/1: the array lengths are unequal" | error
else . as $in
| $integers
| add as $sum
| accumulate as $p
| ($sum|prn + 1) as $random
| $in[first(range(0; $p|length) | select( $random <= $p[.] ))]
end ;
# Input should be a JSON object giving probabilities of each key as a rational: {n, d}
def choose($n):
lcm(.[].d) as $lcm
| ([.[] | $lcm * .n / .d]) as $p
| keys_unsorted as $items
| range(0; $n)
| $items | randomly($p);
# Print a table comparing expected, observed and the ratio
# (expected - observed)^2 / expected
def compare( $expected; $observed ):
def p($n): align_decimal($n) | lpad(8);
" : expected observed (e-o)^2 / e",
( $expected
| keys_unsorted[] as $k
| .[$k] as $e
| ($observed[$k] // 0) as $o
| "\($k|lpad(6)) : \($e|p(1)) \($o|floor|lpad(8)) \( (($e - $o) | (.*.) / $e) | p(2))" );
# The specific task
def probabilities:
{ "aleph": r(1; 5),
"beth": r(1; 6),
"gimel": r(1; 7),
"daleth": r(1; 8),
"he": r(1; 9),
"waw": r(1; 10),
"zayin": r(1; 11),
"heth": r(1759; 27720)
};
def task($n):
probabilities
| bow(choose($n)) as $observed
| compare( map_values($n * .n / .d); $observed ) ;
task(1E6)
'Output
: expected observed (e-o)^2 / e
aleph : 200000.0 200247 0.30
beth : 166666.6 166668 1.06
gimel : 142857.1 142787 0.03
daleth : 125000.0 124789 0.35
he : 111111.1 111280 0.25
waw : 100000.0 100165 0.27
zayin : 90909.0 90716 0.41
heth : 63455.9 63348 0.18
Julia
I made the solution to this task more difficult than I had anticipated by using the Hebrew characters (rather than their anglicised names) as labels for the sampled collection of objects. In doing so, I encountered an interesting subtlety of bidirectional text in Unicode. Namely, that strong right-to-left characters, such as those of Hebrew, override the directionality of European digits, which have weak directionality. Because of this property of Unicode, my table of items and yields had its lines of data interpreted as if it were entirely Hebrew and output in reverse order (from my English speaking perspective). I was able to get the table to display as I liked on my terminal by preceding the the Hebrew characters by the Unicode RLI (right-to-left isolate) control character (U+2067). However, when I pasted this output into this Rosetta Code entry, the display reverted to the "backwards" version. Rather than continue the struggle, trying to force this entry to display as it does on my terminal, I created an alternative version of the table. This "Displayable Here" table adds "yields" to to each line, and this strong left-to-right text makes the whole line display as left-to-right (without the need for a RLI characer).
using Printf
p = [1/i for i in 5:11]
plen = length(p)
q = [0.0, [sum(p[1:i]) for i = 1:plen]]
plab = [char(i) for i in 0x05d0:(0x05d0+plen)]
hi = 10^6
push!(p, 1.0 - sum(p))
plen += 1
accum = zeros(Int, plen)
for i in 1:hi
accum[sum(rand() .>= q)] += 1
end
r = accum/hi
println("Rates at which items are selected (", hi, " trials).")
println(" Item Expected Actual")
for i in 1:plen
println(@sprintf(" \u2067%s %8.6f %8.6f", plab[i], p[i], r[i]))
end
println()
println("Rates at which items are selected (", hi, " trials).")
println(" Item Count Expected Actual")
for i in 1:plen
println(@sprintf(" %s yields %6d %8.6f %8.6f",
plab[i], accum[i], p[i], r[i]))
end
- Output:
Original
This table displays properly on my terminal, but the lines of data are reversed in this display.
Rates at which items are selected (1000000 trials). Item Expected Actual א 0.200000 0.199872 ב 0.166667 0.166618 ג 0.142857 0.143302 ד 0.125000 0.125040 ה 0.111111 0.110602 ו 0.100000 0.099833 ז 0.090909 0.091313 ח 0.063456 0.063420
Displayable Here
This is the same data, less elegantly presented but accurately displayed on both my terminal and here at Rosetta Code.
Rates at which items are selected (1000000 trials). Item Count Expected Actual א yields 199872 0.200000 0.199872 ב yields 166618 0.166667 0.166618 ג yields 143302 0.142857 0.143302 ד yields 125040 0.125000 0.125040 ה yields 110602 0.111111 0.110602 ו yields 99833 0.100000 0.099833 ז yields 91313 0.090909 0.091313 ח yields 63420 0.063456 0.063420
Kotlin
// version 1.0.6
fun main(args: Array<String>) {
val letters = arrayOf("aleph", "beth", "gimel", "daleth", "he", "waw", "zayin", "heth")
val actual = IntArray(8)
val probs = doubleArrayOf(1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0, 0.0)
val cumProbs = DoubleArray(8)
cumProbs[0] = probs[0]
for (i in 1..6) cumProbs[i] = cumProbs[i - 1] + probs[i]
cumProbs[7] = 1.0
probs[7] = 1.0 - cumProbs[6]
val n = 1000000
(1..n).forEach {
val rand = Math.random()
when {
rand <= cumProbs[0] -> actual[0]++
rand <= cumProbs[1] -> actual[1]++
rand <= cumProbs[2] -> actual[2]++
rand <= cumProbs[3] -> actual[3]++
rand <= cumProbs[4] -> actual[4]++
rand <= cumProbs[5] -> actual[5]++
rand <= cumProbs[6] -> actual[6]++
else -> actual[7]++
}
}
var sumActual = 0.0
println("Letter\t Actual Expected")
println("------\t-------- --------")
for (i in 0..7) {
val generated = actual[i].toDouble() / n
println("${letters[i]}\t${String.format("%8.6f %8.6f", generated, probs[i])}")
sumActual += generated
}
println("\t-------- --------")
println("\t${"%8.6f".format(sumActual)} 1.000000")
}
- Output:
Letter Actual Expected
------ -------- --------
aleph 0.199427 0.200000
beth 0.166862 0.166667
gimel 0.142756 0.142857
daleth 0.125442 0.125000
he 0.110868 0.111111
waw 0.100405 0.100000
zayin 0.090799 0.090909
heth 0.063441 0.063456
-------- --------
1.000000 1.000000
Liberty BASIC
names$="aleph beth gimel daleth he waw zayin heth"
dim sum(8)
dim counter(8)
s = 0
for i = 1 to 7
s = s+1/(i+4)
sum(i)=s
next
N =1000000 ' number of throws
for i =1 to N
rand =rnd( 1)
for j = 1 to 7
if sum(j)> rand then exit for
next
counter(j)=counter(j)+1
next
print "Observed", "Intended"
for i = 1 to 8
print word$(names$, i), using( "#.#####", counter(i) /N), using( "#.#####", 1/(i+4))
nextLua
items = {}
items["aleph"] = 1/5.0
items["beth"] = 1/6.0
items["gimel"] = 1/7.0
items["daleth"] = 1/8.0
items["he"] = 1/9.0
items["waw"] = 1/10.0
items["zayin"] = 1/11.0
items["heth"] = 1759/27720
num_trials = 1000000
samples = {}
for item, _ in pairs( items ) do
samples[item] = 0
end
math.randomseed( os.time() )
for i = 1, num_trials do
z = math.random()
for item, _ in pairs( items ) do
if z < items[item] then
samples[item] = samples[item] + 1
break;
else
z = z - items[item]
end
end
end
for item, _ in pairs( items ) do
print( item, samples[item]/num_trials, items[item] )
end
Output
gimel 0.142606 0.14285714285714 heth 0.063434 0.063455988455988 beth 0.166788 0.16666666666667 zayin 0.091097 0.090909090909091 daleth 0.124772 0.125 aleph 0.200541 0.2 he 0.1107 0.11111111111111 waw 0.100062 0.1
Mathematica /Wolfram Language
Built-in function can already do a weighted random choosing. Example for making a million random choices would be:
choices={{"aleph", 1/5},{"beth", 1/6},{"gimel", 1/7},{"daleth", 1/8},{"he", 1/9},{"waw", 1/10},{"zayin", 1/11},{"heth", 1759/27720}};
data=RandomChoice[choices[[All,2]]->choices[[All,1]],10^6];
To compare the data we use the following code to make a table:
Grid[{#[[1]],N[Count[data,#[[1]]]/10^6],N[#[[2]]]}&/@choices]
gives back (item, attained prob., target prob.):
aleph 0.200036 0.2 beth 0.166591 0.166667 gimel 0.142699 0.142857 daleth 0.125018 0.125 he 0.111306 0.111111 waw 0.100433 0.1 zayin 0.090671 0.0909091 heth 0.063246 0.063456
MATLAB
function probChoice
choices = {'aleph' 'beth' 'gimel' 'daleth' 'he' 'waw' 'zayin' 'heth'};
w = [1/5 1/6 1/7 1/8 1/9 1/10 1/11 1759/27720];
R = randsample(length(w), 1e6, true, w);
T = tabulate(R);
fprintf('Value\tCount\tPercent\tGoal\n')
for k = 1:size(T, 1)
fprintf('%6s\t%.f\t%.2f%%\t%.2f%%\n', ...
choices{k}, T(k, 2), T(k, 3), 100*w(k))
end
end
- Output:
Value Count Percent Goal
aleph 199635 19.96% 20.00%
beth 166427 16.64% 16.67%
gimel 143342 14.33% 14.29%
daleth 125014 12.50% 12.50%
he 111031 11.10% 11.11%
waw 99920 9.99% 10.00%
zayin 91460 9.15% 9.09%
heth 63171 6.32% 6.35%
function probChoice
choices = {'aleph' 'beth' 'gimel' 'daleth' 'he' 'waw' 'zayin' 'heth'};
w = [1/5 1/6 1/7 1/8 1/9 1/10 1/11 1759/27720];
nSamp = 1e6;
nChoice = length(w);
R = rand(nSamp, 1);
wCS = cumsum(w);
results = zeros(1, nChoice);
fprintf('Value\tCount\tPercent\tGoal\n')
for k = 1:nChoice
choiceKIdxs = R < wCS(k);
R(choiceKIdxs) = k;
results(k) = sum(choiceKIdxs);
fprintf('%6s\t%.f\t%.2f%%\t%.2f%%\n', ...
choices{k}, results(k), 100*results(k)/nSamp, 100*w(k))
end
end
- Output:
Value Count Percent Goal
aleph 200327 20.03% 20.00%
beth 166318 16.63% 16.67%
gimel 143040 14.30% 14.29%
daleth 125136 12.51% 12.50%
he 111251 11.13% 11.11%
waw 99946 9.99% 10.00%
zayin 90974 9.10% 9.09%
heth 63008 6.30% 6.35%
Nim
import tables, random, strformat, times
var start = cpuTime()
const
NumTrials = 1_000_000
Probabilities = {"aleph": 1 / 5, "beth": 1 / 6, "gimel": 1 / 7, "daleth": 1 / 8,
"he": 1 / 9, "waw": 1 / 10, "zayin": 1 / 11, "heth": 1759 / 27720}.toTable
var samples: CountTable[string]
randomize()
for i in 1 .. NumTrials:
var z = rand(1.0)
for item, prob in Probabilities.pairs:
if z < prob:
samples.inc(item)
break
else:
z -= prob
var s1, s2 = 0.0
echo " Item Target Results Differences"
echo "====== ======== ======== ==========="
for item, prob in Probabilities.pairs:
let r = samples[item] / NumTrials
s1 += r * 100
s2 += prob * 100
echo &"{item:<6} {prob:.6f} {r:.6f} {100 * (1 - r / prob):9.6f} %"
echo "====== ======== ======== "
echo &"Total: {s2:^8.2f} {s1:^8.2f}"
echo &"\nExecution time: {cpuTime()-start:.2f} s"
- Output:
Item Target Results Differences ====== ======== ======== =========== he 0.111111 0.111485 -0.336500 % heth 0.063456 0.063758 -0.475939 % zayin 0.090909 0.090811 0.107900 % aleph 0.200000 0.199603 0.198500 % gimel 0.142857 0.142702 0.108600 % daleth 0.125000 0.124831 0.135200 % beth 0.166667 0.167160 -0.296000 % waw 0.100000 0.099650 0.350000 % ====== ======== ======== Total: 100.00 100.00 Execution time: 0.05 s
OCaml
let p = [
"Aleph", 1.0 /. 5.0;
"Beth", 1.0 /. 6.0;
"Gimel", 1.0 /. 7.0;
"Daleth", 1.0 /. 8.0;
"He", 1.0 /. 9.0;
"Waw", 1.0 /. 10.0;
"Zayin", 1.0 /. 11.0;
"Heth", 1759.0 /. 27720.0;
]
let rec take k = function
| (v, p)::tl -> if k < p then v else take (k -. p) tl
| _ -> invalid_arg "take"
let () =
let n = 1_000_000 in
Random.self_init();
let h = Hashtbl.create 3 in
List.iter (fun (v, _) -> Hashtbl.add h v 0) p;
let tot = List.fold_left (fun acc (_, p) -> acc +. p) 0.0 p in
for i = 1 to n do
let sel = take (Random.float tot) p in
let n = Hashtbl.find h sel in
Hashtbl.replace h sel (succ n) (* count the number of each item *)
done;
List.iter (fun (v, p) ->
let d = Hashtbl.find h v in
Printf.printf "%s \t %f %f\n" v p (float d /. float n)
) p
Output:
Aleph 0.200000 0.200272 Beth 0.166667 0.166381 Gimel 0.142857 0.142497 Daleth 0.125000 0.125005 He 0.111111 0.111272 Waw 0.100000 0.100069 Zayin 0.090909 0.091136 Heth 0.063456 0.063368
PARI/GP
pc()={
my(v=[5544,10164,14124,17589,20669,23441,25961,27720],u=vector(8),e);
for(i=1,1e6,
my(r=random(27720));
for(j=1,8,
if(r<v[j], u[j]++; break)
)
);
e=precision([1/5,1/6,1/7,1/8,1/9,1/10,1/11,1759/27720]*1e6,9); \\ truncate to 9 decimal places
print("Totals: "u);
print("Expected: "e);
print("Diff: ",u-e);
print("StDev: ",vector(8,i,sqrt(abs(u[i]-v[i])/e[i])));
};Pascal
Free Pascal
Creating an array of length of divisor.Only useful for a small divisor.
program Probablistic;
const
rounds = 1000*1000;
names : array[0..7] of string =('aleph','beth','gimel','daleth','he','waw','zayin','heth');
quotient = 5*6*7*8*9*10*11;
RezValues: array[0..7] of integer=(5,6,7,8,9,10,11,0);
//remaining heth = 27720/1759
function GCD(a, b: Int64): Int64;
var
temp: Int64;
begin
while b <> 0 do
begin
temp := b;
b := a mod b;
a := temp
end;
Exit(a);
end;
var
cnt : array of Uint32;
i ,z,n,tmp,idx,lmt : NativeInt;
sumTot : NativeUint;
begin
randomize;
z := 0;
n := 1;
For i := 0 to 6 do
begin
z := z*RezValues[i]+n;
n := n*RezValues[i];
tmp := GCD(z,n);
z := z div tmp;
n := n div tmp;
end;
//create the values
setlength(cnt,n+1);
For i := 1 to rounds do
inc(cnt[trunc(random()*n)]);
//count occurences in range
writeln('Item':9,'expected':10,'actual':10);
sumTot := 0;
idx := 0;
lmt := 0;
For i := 0 to 6 do
begin
lmt += n DIV RezValues[i];
tmp := 0;
repeat
inc(tmp,cnt[idx]);
inc(idx);
until idx >= lmt;
sumTot += tmp;
writeln(names[i]:9,1/RezValues[i]:10:6,tmp/rounds:10:6);
end;
//the remaining for heth
writeln(names[7]:9,(n-z)/n:10:6,1-sumTot/rounds:10:6);
end.
- Output:
Item expected actual
aleph 0.200000 0.200282
beth 0.166667 0.166672
gimel 0.142857 0.142627
daleth 0.125000 0.124634
he 0.111111 0.111547
waw 0.100000 0.100164
zayin 0.090909 0.090630
heth 0.063456 0.063444
Perl
use List::Util qw(first sum);
use constant TRIALS => 1e6;
sub prob_choice_picker {
my %options = @_;
my ($n, @a) = 0;
while (my ($k,$v) = each %options) {
$n += $v;
push @a, [$n, $k];
}
return sub {
my $r = rand;
( first {$r <= $_->[0]} @a )->[1];
};
}
my %ps =
(aleph => 1/5,
beth => 1/6,
gimel => 1/7,
daleth => 1/8,
he => 1/9,
waw => 1/10,
zayin => 1/11);
$ps{heth} = 1 - sum values %ps;
my $picker = prob_choice_picker %ps;
my %results;
for (my $n = 0 ; $n < TRIALS ; ++$n) {
++$results{$picker->()};
}
print "Event Occurred Expected Difference\n";
foreach (sort {$results{$b} <=> $results{$a}} keys %results) {
printf "%-6s %f %f %f\n",
$_, $results{$_}/TRIALS, $ps{$_},
abs($results{$_}/TRIALS - $ps{$_});
}
Sample output:
Event Occurred Expected Difference aleph 0.198915 0.200000 0.001085 beth 0.166804 0.166667 0.000137 gimel 0.142992 0.142857 0.000135 daleth 0.125155 0.125000 0.000155 he 0.111160 0.111111 0.000049 waw 0.100229 0.100000 0.000229 zayin 0.091014 0.090909 0.000105 heth 0.063731 0.063456 0.000275
Phix
with javascript_semantics constant lim = 1000000, {names, probs} = columnize({{"aleph", 1/5}, {"beth", 1/6}, {"gimel", 1/7}, {"daleth", 1/8}, {"he", 1/9}, {"waw", 1/10}, {"zayin", 1/11}, {"heth", 1759/27720}}) sequence results = repeat(0,length(names)) for j=1 to lim do atom r = rnd() for i=1 to length(probs) do r -= probs[i] if r<=0 then results[i]+=1 exit end if end for end for printf(1," Name Actual Expected\n") for i=1 to length(probs) do printf(1,"%6s %8.6f %8.6f\n",{names[i],results[i]/lim,probs[i]}) end for
- Output:
Name Actual Expected
aleph 0.201010 0.200000
beth 0.166311 0.166667
gimel 0.143354 0.142857
daleth 0.124841 0.125000
he 0.110544 0.111111
waw 0.100228 0.100000
zayin 0.090270 0.090909
heth 0.063442 0.063456
Phixmonti
/# Rosetta Code problem: http://rosettacode.org/wiki/Probabilistic_choice
by Galileo, 05/2022 #/
include ..\Utilitys.pmt
( ( "aleph" 0.200000 0 ) ( "beth" 0.166667 0 ) ( "gimel" 0.142857 0 ) ( "daleth" 0.125000 0 )
( "he" 0.111111 0 ) ( "waw" 0.100000 0 ) ( "zayin" 0.090909 0 ) ( "heth" 0.063456 0 ) )
len 1 swap 2 tolist var lprob
1000000 var trial
trial for drop
rand >ps
0 >ps
lprob for var i
( i 2 ) sget ps> +
tps swap dup >ps < if
( i 3 ) sget 1 + ( i 3 ) sset
exitfor
endif
endfor
ps> ps> drop drop
endfor
( "item" "\t" "actual" "\t\t" "theoretical" ) lprint nl nl
lprob for drop
pop swap
1 get "\t" rot 3 get trial / "\t" rot 2 get nip "\n" 6 tolist lprint
endfor- Output:
item actual theoretical aleph 0.200018 0.2 beth 0.166987 0.166667 gimel 0.142654 0.142857 daleth 0.125217 0.125 he 0.111668 0.111111 waw 0.099827 0.1 zayin 0.090311 0.090909 heth 0.063318 0.063456 === Press any key to exit ===
PicoLisp
(let (Count 1000000 Denom 27720 N Denom)
(let Probs
(mapcar
'((I S)
(prog1 (cons N (*/ Count I) 0 S)
(dec 'N (/ Denom I)) ) )
(range 5 12)
'(aleph beth gimel daleth he waw zayin heth) )
(do Count
(inc (cddr (rank (rand 1 Denom) Probs T))) )
(let Fmt (-6 12 12)
(tab Fmt NIL "Probability" "Result")
(for X Probs
(tab Fmt
(cdddr X)
(format (cadr X) 6)
(format (caddr X) 6) ) ) ) ) )Output:
Probability Result aleph 0.200000 0.199760 beth 0.166667 0.166878 gimel 0.142857 0.142977 daleth 0.125000 0.124983 he 0.111111 0.111200 waw 0.100000 0.100173 zayin 0.090909 0.090591 heth 0.083333 0.063438
PL/I
probch: Proc Options(main);
Dcl prob(8) Dec Float(15) Init((1/5.0), /* aleph */
(1/6.0), /* beth */
(1/7.0), /* gimel */
(1/8.0), /* daleth */
(1/9.0), /* he */
(1/10.0), /* waw */
(1/11.0), /* zayin */
(1759/27720));/* heth */
Dcl what(8) Char(6) Init('aleph ','beth ','gimel ','daleth',
'he ','waw ','zayin ','heth ');
Dcl ulim(0:8) Dec Float(15) Init((9)0);
Dcl i Bin Fixed(31);
Dcl ifloat Dec Float(15);
Dcl one Dec Float(15) Init(1);
Dcl num Dec Float(15) Init(1759);
Dcl denom Dec Float(15) Init(27720);
Dcl x Dec Float(15) Init(0);
Dcl pr Dec Float(15) Init(0);
Dcl (n,nn) Bin Fixed(31);
Dcl cnt(8) Bin Fixed(31) Init((8)0);
nn=1000000;
Do i=1 To 8;
ifloat=i+4;
If i<8 Then
prob(i)=one/ifloat;
Else
prob(i)=num/denom;
Ulim(i)=ulim(i-1)+prob(i);
/* Put Skip list(i,prob(i),ulim(i));*/
End;
Do n=1 To nn;
x=random();
Do i=1 To 8;
If x<ulim(i) Then Leave;
End;
cnt(i)+=1;
End;
Put Edit('letter occurs frequency expected ')(Skip,a);
Put Edit('------ ------ ---------- ----------')(Skip,a);
Do i=1 To 8;
pr=float(cnt(i))/float(nn);
Put Edit(what(i),cnt(i),pr,prob(i))(Skip,a,f(10),x(2),2(f(11,8)));
End;
End;- Output:
One million trials letter occurs frequency expected ------ ------ ---------- --------- aleph 199989 0.19998900 0.20000000 beth 167338 0.16733800 0.16666667 gimel 142968 0.14296800 0.14285714 daleth 124840 0.12484000 0.12500000 he 110620 0.11062000 0.11111111 waw 99744 0.09974400 0.10000000 zayin 90930 0.09093000 0.09090909 heth 63571 0.06357100 0.06345599 One hundred million trials letter occurs frequency expected ------ ------ ---------- ---------- aleph 20002222 0.20002222 0.20000000 beth 16665226 0.16665226 0.16666667 gimel 14289674 0.14289674 0.14285714 daleth 12498182 0.12498182 0.12500000 he 11108704 0.11108704 0.11111111 waw 10002442 0.10002442 0.10000000 zayin 9087412 0.09087412 0.09090909 heth 6346138 0.06346138 0.06345599
PowerShell
The guts of this script are translated from the Java Script entry. Then I stole the idea to show the actual Hebrew character from Julia.
$character = [PSCustomObject]@{
aleph = [PSCustomObject]@{Expected=1/5 ; Alpha="א"}
beth = [PSCustomObject]@{Expected=1/6 ; Alpha="ב"}
gimel = [PSCustomObject]@{Expected=1/7 ; Alpha="ג"}
daleth = [PSCustomObject]@{Expected=1/8 ; Alpha="ד"}
he = [PSCustomObject]@{Expected=1/9 ; Alpha="ה"}
waw = [PSCustomObject]@{Expected=1/10 ; Alpha="ו"}
zayin = [PSCustomObject]@{Expected=1/11 ; Alpha="ז"}
heth = [PSCustomObject]@{Expected=1759/27720; Alpha="ח"}
}
$sum = 0
$iterations = 1000000
$cumulative = [ordered]@{}
$randomly = [ordered]@{}
foreach ($name in $character.PSObject.Properties.Name)
{
$sum += $character.$name.Expected
$cumulative.$name = $sum
$randomly.$name = 0
}
for ($i = 0; $i -lt $iterations; $i++)
{
$random = Get-Random -Minimum 0.0 -Maximum 1.0
foreach ($name in $cumulative.Keys)
{
if ($random -le $cumulative.$name)
{
$randomly.$name++
break
}
}
}
foreach ($name in $character.PSObject.Properties.Name)
{
[PSCustomObject]@{
Name = $name
Expected = $character.$name.Expected
Actual = $randomly.$name / $iterations
Character = $character.$name.Alpha
}
}
- Output:
Name Expected Actual Character ---- -------- ------ --------- aleph 0.2 0.199823 א beth 0.166666666666667 0.166744 ב gimel 0.142857142857143 0.143312 ג daleth 0.125 0.125153 ד he 0.111111111111111 0.110984 ה waw 0.1 0.099667 ו zayin 0.0909090909090909 0.091135 ז heth 0.0634559884559885 0.063182 ח
PureBasic
#times=1000000
Structure Item
name.s
prob.d
Amount.i
EndStructure
If OpenConsole()
Define i, j, d.d, e.d, txt.s
Dim Mapps.Item(7)
Mapps(0)\name="aleph": Mapps(0)\prob=1/5.0
Mapps(1)\name="beth": Mapps(1)\prob=1/6.0
Mapps(2)\name="gimel": Mapps(2)\prob=1/7.0
Mapps(3)\name="daleth":Mapps(3)\prob=1/8.0
Mapps(4)\name="he": Mapps(4)\prob=1/9.0
Mapps(5)\name="waw": Mapps(5)\prob=1/10.0
Mapps(6)\name="zayin": Mapps(6)\prob=1/11.0
Mapps(7)\name="heth": Mapps(7)\prob=1759/27720.0
For i=1 To #times
d=Random(#MAXLONG)/#MAXLONG ; Get a random number
e=0.0
For j=0 To ArraySize(Mapps())
e+Mapps(j)\prob ; Get span for current itme
If d<=e ; Check if it is within this span?
Mapps(j)\Amount+1 ; If so, count it.
Break
EndIf
Next j
Next i
PrintN("Sample times: "+Str(#times)+#CRLF$)
For j=0 To ArraySize(Mapps())
d=Mapps(j)\Amount/#times
txt=LSet(Mapps(j)\name,7)+" should be "+StrD(Mapps(j)\prob)+" is "+StrD(d)
PrintN(txt+" | Deviatation "+RSet(StrD(100.0-100.0*Mapps(j)\prob/d,3),6)+"%")
Next
Print(#CRLF$+"Press ENTER to exit"):Input()
CloseConsole()
EndIf
Output may look like
Sample times: 1000000 aleph should be 0.2000000000 is 0.1995520000 | Deviatation -0.225% beth should be 0.1666666667 is 0.1673270000 | Deviatation 0.395% gimel should be 0.1428571429 is 0.1432040000 | Deviatation 0.242% daleth should be 0.1250000000 is 0.1251850000 | Deviatation 0.148% he should be 0.1111111111 is 0.1109550000 | Deviatation -0.141% waw should be 0.1000000000 is 0.0999220000 | Deviatation -0.078% zayin should be 0.0909090909 is 0.0902240000 | Deviatation -0.759% heth should be 0.0634559885 is 0.0636310000 | Deviatation 0.275% Press ENTER to exit
Python
Two different algorithms are coded.
import random, bisect
def probchoice(items, probs):
'''\
Splits the interval 0.0-1.0 in proportion to probs
then finds where each random.random() choice lies
'''
prob_accumulator = 0
accumulator = []
for p in probs:
prob_accumulator += p
accumulator.append(prob_accumulator)
while True:
r = random.random()
yield items[bisect.bisect(accumulator, r)]
def probchoice2(items, probs, bincount=10000):
'''\
Puts items in bins in proportion to probs
then uses random.choice() to select items.
Larger bincount for more memory use but
higher accuracy (on avarage).
'''
bins = []
for item,prob in zip(items, probs):
bins += [item]*int(bincount*prob)
while True:
yield random.choice(bins)
def tester(func=probchoice, items='good bad ugly'.split(),
probs=[0.5, 0.3, 0.2],
trials = 100000
):
def problist2string(probs):
'''\
Turns a list of probabilities into a string
Also rounds FP values
'''
return ",".join('%8.6f' % (p,) for p in probs)
from collections import defaultdict
counter = defaultdict(int)
it = func(items, probs)
for dummy in xrange(trials):
counter[it.next()] += 1
print "\n##\n## %s\n##" % func.func_name.upper()
print "Trials: ", trials
print "Items: ", ' '.join(items)
print "Target probability: ", problist2string(probs)
print "Attained probability:", problist2string(
counter[x]/float(trials) for x in items)
if __name__ == '__main__':
items = 'aleph beth gimel daleth he waw zayin heth'.split()
probs = [1/(float(n)+5) for n in range(len(items))]
probs[-1] = 1-sum(probs[:-1])
tester(probchoice, items, probs, 1000000)
tester(probchoice2, items, probs, 1000000)
Sample output:
## ## PROBCHOICE ## Trials: 1000000 Items: aleph beth gimel daleth he waw zayin heth Target probability: 0.200000,0.166667,0.142857,0.125000,0.111111,0.100000,0.090909,0.063456 Attained probability: 0.200050,0.167109,0.143364,0.124690,0.111237,0.099661,0.090338,0.063551 ## ## PROBCHOICE2 ## Trials: 1000000 Items: aleph beth gimel daleth he waw zayin heth Target probability: 0.200000,0.166667,0.142857,0.125000,0.111111,0.100000,0.090909,0.063456 Attained probability: 0.199720,0.166424,0.142474,0.124561,0.111511,0.100313,0.091316,0.063681
Quackery
Uses point$ from the bignum rational arithmetic module bigrat.qky. 10 point$ returns a ratio as a decimal string accurate to 10 decimal places with round-to-nearest on the final digit.
[ $ "bigrat.qky" loadfile ] now!
( --------------- zen object orientation -------------- )
[ immovable
]this[ swap do ]done[ ] is object ( [ --> )
[ ]'[ ] is method ( --> [ )
[ method
[ dup share
swap put ] ] is localise ( --> )
[ method [ release ] ] is delocalise ( --> )
( ------------------ rand-gen methods ----------------- )
[ method
[ dup take
2 split drop
' [ 0 0 ] join
swap put ] ] is reset-gen ( --> [ )
[ method
[ dup take
dup 2 peek 1+
swap 2 poke
dup 1 peek random
over 0 peek <
if
[ dup 3 peek 1+
swap 3 poke ]
swap put ] ] is rand-gen ( --> [ )
[ method
[ dup echo say ": "
share
dup 2 peek dup echo
say " trials" cr
say " Actual: "
over 3 peek
swap 10 point$ echo$ cr
say " Expected: "
dup 0 peek
swap 1 peek
10 point$ echo$ cr
cr ] ] is report ( --> [ )
( ------------------ rand-gen objects ----------------- )
[ object [ 1 5 0 0 ] ] is aleph ( [ --> )
[ object [ 1 6 0 0 ] ] is beth ( [ --> )
[ object [ 1 7 0 0 ] ] is gimel ( [ --> )
[ object [ 1 8 0 0 ] ] is daleth ( [ --> )
[ object [ 1 9 0 0 ] ] is he ( [ --> )
[ object [ 1 10 0 0 ] ] is waw ( [ --> )
[ object [ 1 11 0 0 ] ] is zayin ( [ --> )
[ object [ 1759 27720 0 0 ] ] is heth ( [ --> )
' [ aleph beth gimel daleth he waw zayin heth ]
dup witheach [ reset-gen swap do ]
dup witheach
[ 1000000 times
[ rand-gen over do ]
drop ]
witheach [ report swap do ]- Output:
aleph: 1000000 trials
Actual: 0.199454
Expected: 0.2
beth: 1000000 trials
Actual: 0.166854
Expected: 0.1666666667
gimel: 1000000 trials
Actual: 0.142329
Expected: 0.1428571429
daleth: 1000000 trials
Actual: 0.12457
Expected: 0.125
he: 1000000 trials
Actual: 0.111143
Expected: 0.1111111111
waw: 1000000 trials
Actual: 0.100061
Expected: 0.1
zayin: 1000000 trials
Actual: 0.090979
Expected: 0.0909090909
heth: 1000000 trials
Actual: 0.063131
Expected: 0.0634559885
R
prob = c(aleph=1/5, beth=1/6, gimel=1/7, daleth=1/8, he=1/9, waw=1/10, zayin=1/11, heth=1759/27720)
# Note that R doesn't actually require the weights
# vector for rmultinom to sum to 1.
hebrew = c(rmultinom(1, 1e6, prob))
d = data.frame(
Requested = prob,
Obtained = hebrew/sum(hebrew))
print(d)
Sample output:
Requested Obtained aleph 0.20000000 0.200311 beth 0.16666667 0.167160 gimel 0.14285714 0.141997 daleth 0.12500000 0.124644 he 0.11111111 0.110984 waw 0.10000000 0.099927 zayin 0.09090909 0.091365 heth 0.06345599 0.063612
A histogram of the data is also possible using, for example,
library(ggplot2)
qplot(factor(names(prob), levels = names(prob)), hebrew, geom = "histogram")
Racket
probabalistic-choice uses inexact (float) arithmetic
probabalistic-choice/exact uses fractions and greatest common denominators and the likes
The test submodule is used for unit tests, and is not run when this code is loaded as a module. Either run the program in DrRacket or run `raco test prob-choice.rkt`
#lang racket
;;; returns a probabalistic choice from the sequence choices
;;; choices generates two values -- the chosen value and a
;;; probability (weight) of the choice.
;;;
;;; Note that a hash where keys are choices and values are probabilities
;;; is such a sequence.
;;;
;;; if the total probability < 1 then choice could return #f
;;; if the total probability > 1 then some choices may be impossible
(define (probabalistic-choice choices)
(let-values
(((_ choice) ;; the fold provides two values, we only need the second
;; the first will always be a negative number showing that
;; I've run out of random steam
(for/fold
((rnd (random))
(choice #f))
(((v p) choices)
#:break (<= rnd 0))
(values (- rnd p) v))))
choice))
;;; ditto, but all probabilities must be exact rationals
;;; the optional lcd
;;;
;;; not the most efficient, since it provides a wrapper (and demo)
;;; for p-c/i-w below
(define (probabalistic-choice/exact
choices
#:gcd (GCD (/ (apply gcd (hash-values choices)))))
(probabalistic-choice/integer-weights
(for/hash (((k v) choices))
(values k (* v GCD)))
#:sum-of-weights GCD))
;;; this proves useful in Rock-Paper-Scissors
(define (probabalistic-choice/integer-weights
choices
#:sum-of-weights (sum-of-weights (apply + (hash-values choices))))
(let-values
(((_ choice)
(for/fold
((rnd (random sum-of-weights))
(choice #f))
(((v p) choices)
#:break (< rnd 0))
(values (- rnd p) v))))
choice))
(module+ test
(define test-samples (make-parameter 1000000))
(define (test-p-c-function f w)
(define test-selection (make-hash))
(for* ((i (in-range 0 (test-samples)))
(c (in-value (f w))))
(when (zero? (modulo i 100000)) (eprintf "~a," (quotient i 100000)))
(hash-update! test-selection c add1 0))
(printf "~a~%choice\tcount\texpected\tratio\terror~%" f)
(for* (((k v) (in-hash test-selection))
(e (in-value (* (test-samples) (hash-ref w k)))))
(printf "~a\t~a\t~a\t~a\t~a%~%"
k v e
(/ v (test-samples))
(real->decimal-string
(exact->inexact (* 100 (/ (- v e) e)))))))
(define test-weightings/rosetta
(hash
'aleph 1/5
'beth 1/6
'gimel 1/7
'daleth 1/8
'he 1/9
'waw 1/10
'zayin 1/11
'heth 1759/27720; adjusted so that probabilities add to 1
))
(define test-weightings/50:50 (hash 'woo 1/2 'yay 1/2))
(define test-weightings/1:2:3 (hash 'woo 1 'yay 2 'foo 3))
(test-p-c-function probabalistic-choice test-weightings/50:50)
(test-p-c-function probabalistic-choice/exact test-weightings/50:50)
(test-p-c-function probabalistic-choice test-weightings/rosetta)
(test-p-c-function probabalistic-choice/exact test-weightings/rosetta))
Output (note that the progress counts, which go to standard error, are interleaved with the output on standard out)
0,1,2,3,4,5,6,7,8,9,#<procedure:probabalistic-choice> choice count expected ratio error yay 499744 500000 15617/31250 -0.05% woo 500256 500000 15633/31250 0.05% 0,1,2,3,4,5,6,7,8,9,#<procedure:probabalistic-choice/exact> choice count expected ratio error yay 499852 500000 124963/250000 -0.03% woo 500148 500000 125037/250000 0.03% 0,1,2,3,4,5,6,7,8,9,#<procedure:probabalistic-choice> choice count expected ratio error daleth 124964 125000 31241/250000 -0.03% zayin 90233 1000000/11 90233/1000000 -0.74% gimel 142494 1000000/7 71247/500000 -0.25% heth 64045 43975000/693 12809/200000 0.93% aleph 199690 200000 19969/100000 -0.15% beth 166861 500000/3 166861/1000000 0.12% waw 100075 100000 4003/40000 0.07% he 111638 1000000/9 55819/500000 0.47% 0,1,2,3,4,5,6,7,8,9,#<procedure:probabalistic-choice/exact> choice count expected ratio error beth 166423 500000/3 166423/1000000 -0.15% heth 63462 43975000/693 31731/500000 0.01% daleth 125091 125000 125091/1000000 0.07% waw 99820 100000 4991/50000 -0.18% aleph 200669 200000 200669/1000000 0.33% gimel 142782 1000000/7 71391/500000 -0.05% zayin 90478 1000000/11 45239/500000 -0.47% he 111275 1000000/9 4451/40000 0.15%
Raku
(formerly Perl 6)
constant TRIALS = 1e6;
constant @event = <aleph beth gimel daleth he waw zayin heth>;
constant @P = flat (1 X/ 5 .. 11), 1759/27720;
constant @cP = [\+] @P;
my atomicint @results[+@event];
(^TRIALS).race.map: { @results[ @cP.first: { $_ > once rand }, :k ]⚛++; }
say 'Event Occurred Expected Difference';
for ^@results {
my ($occurred, $expected) = @results[$_], @P[$_] * TRIALS;
printf "%-9s%8.0f%9.1f%12.1f\n",
@event[$_],
$occurred,
$expected,
abs $occurred - $expected;
}
- Output:
Event Occurred Expected Difference aleph 200369 200000.0 369.0 beth 167005 166666.7 338.3 gimel 142690 142857.1 167.1 daleth 125061 125000.0 61.0 he 110563 111111.1 548.1 waw 100214 100000.0 214.0 zayin 90617 90909.1 292.1 heth 63481 63456.0 25.0
ReScript
let p = [
("Aleph", 1.0 /. 5.0),
("Beth", 1.0 /. 6.0),
("Gimel", 1.0 /. 7.0),
("Daleth", 1.0 /. 8.0),
("He", 1.0 /. 9.0),
("Waw", 1.0 /. 10.0),
("Zayin", 1.0 /. 11.0),
("Heth", 1759.0 /. 27720.0),
]
let prob_take = (arr, k) => {
let rec aux = (i, k) => {
let (v, p) = arr[i]
if k < p { v } else { aux(i+1, (k -. p)) }
}
aux(0, k)
}
{
let n = 1_000_000
let h = Belt.HashMap.String.make(~hintSize=10)
Js.Array2.forEach(p, ((v, _)) =>
Belt.HashMap.String.set(h, v, 0)
)
let tot = Js.Array2.reduce(p, (acc, (_, prob)) => acc +. prob, 0.0)
for _ in 1 to n {
let sel = prob_take(p, tot *. Js.Math.random())
let _n = Belt.HashMap.String.get(h, sel)
let n = Belt.Option.getExn(_n)
Belt.HashMap.String.set(h, sel, (n+1)) /* count the number of each item */
}
Printf.printf("Event expected occurred\n")
Js.Array2.forEach(p, ((v, p)) => {
let _d = Belt.HashMap.String.get(h, v)
let d = Belt.Option.getExn(_d)
Printf.printf("%s \t %8.5g %8.5g\n", v, p, float(d) /. float(n))
}
)
}- Output:
Event expected occurred Aleph 0.2 0.20042 Beth 0.16667 0.16606 Gimel 0.14286 0.14324 Daleth 0.125 0.1252 He 0.11111 0.11085 Waw 0.1 0.099557 Zayin 0.090909 0.090877 Heth 0.063456 0.0638
REXX
Note: REXX can generate random numbers up to a range of 100,000.
A little extra REXX code was added to provide head and foot titles along with the totals.
/*REXX program displays results of probabilistic choices, gen random #s per probability.*/
parse arg trials digs seed . /*obtain the optional arguments from CL*/
if trials=='' | trials=="," then trials= +1e6 /*Not specified? Then use the default.*/
if digs=='' | digs=="," then digs= 15 /* " " " " " " */
if datatype(seed, 'W') then call random ,,seed /*allows repeatability for RANDOM nums.*/
numeric digits digs /*use a specific number of decimal digs*/
names= 'aleph beth gimel daleth he waw zayin heth ───totals───►' /*names of the cells.*/
hi= 100000 /*max REXX RANDOM num*/
z= words(names); #= z - 1 /*#: the number of actual/usable names.*/
$= 0 /*initialize sum of the probabilities. */
do n=1 for #; prob.n= 1 / (n+4); if n==# then prob.n= 1759 / 27720
$= $ + prob.n; Hprob.n= prob.n * hi /*spread the range of probabilities. */
end /*n*/
prob.z= $ /*define the value of the ───totals───.*/
@.= 0 /*initialize all counters in the range.*/
@.z= trials /*define the last counter of " " */
do j=1 for trials; r= random(hi) /*gen TRIAL number of random numbers.*/
do k=1 for # /*for each cell, compute percentages. */
if r<=Hprob.k then @.k= @.k + 1 /* " " " range, bump the counter*/
end /*k*/
end /*j*/
_= '═' /*_: padding used by the CENTER BIF.*/
w= digs + 6 /*W: display width for the percentages*/
d= 4 + max( length(trials), length('count') ) /* [↓] display a formatted top header.*/
say center('name',15,_) center('count',d,_) center('target %',w,_) center('actual %',w,_)
do cell=1 for z /*display each of the cells and totals.*/
say ' ' left( word(names, cell), 13) right(@.cell, d-2) " " ,
left( format( prob.cell * 100, d), w-2) ,
left( format( @.cell/trials * 100, d), w-2) /* [↓] foot title. [↓] */
if cell==# then say center(_,15,_) center(_,d,_) center(_,w,_) center(_,w,_)
end /*c*/ /*stick a fork in it, we are all done.*/
- output when using the default input:
═════name══════ ═══count═══ ══════target %═══════ ══════actual %═══════ aleph 200135 20 20.0135 beth 166912 16.6666666 16.6912 gimel 143222 14.2857142 14.3222 daleth 124991 12.5 12.4991 he 111259 11.1111111 11.1259 waw 100049 10 10.0049 zayin 90978 9.0909090 9.0978 heth 63278 6.3455988 6.3278 ═══════════════ ═══════════ ═════════════════════ ═════════════════════ ───totals───► 1000000 100 100
Ring
# Project : Probabilistic choice
cnt = list(8)
item = ["aleph","beth","gimel","daleth","he","waw","zayin","heth"]
prob = [1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0, 1759/27720]
for trial = 1 to 1000000
r = random(10)/10
p = 0
for i = 1 to len(prob)
p = p + prob[i]
if r < p
cnt[i] = cnt[i] + 1
loop
ok
next
next
see "item actual theoretical" + nl
for i = 1 to len(item)
see "" + item[i] + " " + cnt[i]/1000000 + " " + prob[i] + nl
nextOutput:
item actual theoretical aleph 0.091307 0.200000 beth 0.181073 0.166667 gimel 0.181884 0.142857 daleth 0.090985 0.125000 he 0.090958 0.111111 waw 0.091064 0.100000 zayin 0.091061 0.090909 heth 0 0.063456
RPL
« @ create { aleph..heth } mapping { 5 6 7 8 9 10 11 } INV 1 OVER ∑LIST - + "expected" →TAG @ cumulate odds { } → cumulodds « DUP REVLIST OBJ→ 0 1 ROT START + 'cumulodds' OVER STO+ NEXT @ initialize counters DROP DUP 1 « DROP 0 » DOLIST @ count 1 1000000 START cumulodds RAND ≥ 1 POS DUP2 GET 1 + PUT NEXT @ display results DUP ∑LIST / "occurred" →TAG DUP2 - ABS « MAX » STREAM "max gap" →TAG » » 'TASK' STO
- Output:
3: expected: { .2 .166666666667 .142857142857 .125 .111111111111 .1 9.09090909091E-2 .063455988456 }
2: occurred: { .200228 .16701 .143054 .125387 .111077 .099929 .090411 .062904 }
1: max gap: .000551988456
Ruby
probabilities = {
"aleph" => 1/5.0,
"beth" => 1/6.0,
"gimel" => 1/7.0,
"daleth" => 1/8.0,
"he" => 1/9.0,
"waw" => 1/10.0,
"zayin" => 1/11.0,
}
probabilities["heth"] = 1.0 - probabilities.each_value.inject(:+)
ordered_keys = probabilities.keys
sum, sums = 0.0, {}
ordered_keys.each do |key|
sum += probabilities[key]
sums[key] = sum
end
actual = Hash.new(0)
samples = 1_000_000
samples.times do
r = rand
for k in ordered_keys
if r < sums[k]
actual[k] += 1
break
end
end
end
puts "key expected actual diff"
for k in ordered_keys
act = Float(actual[k]) / samples
val = probabilities[k]
printf "%-8s%.8f %.8f %6.3f %%\n", k, val, act, 100*(act-val)/val
end
- Output:
key expected actual diff aleph 0.20000000 0.19949200 -0.254 % beth 0.16666667 0.16689900 0.139 % gimel 0.14285714 0.14309300 0.165 % daleth 0.12500000 0.12494200 -0.046 % he 0.11111111 0.11037800 -0.660 % waw 0.10000000 0.10030100 0.301 % zayin 0.09090909 0.09162700 0.790 % heth 0.06345599 0.06326800 -0.296 %
Rust
extern crate rand;
use rand::distributions::{IndependentSample, Sample, Weighted, WeightedChoice};
use rand::{weak_rng, Rng};
const DATA: [(&str, f64); 8] = [
("aleph", 1.0 / 5.0),
("beth", 1.0 / 6.0),
("gimel", 1.0 / 7.0),
("daleth", 1.0 / 8.0),
("he", 1.0 / 9.0),
("waw", 1.0 / 10.0),
("zayin", 1.0 / 11.0),
("heth", 1759.0 / 27720.0),
];
const SAMPLES: usize = 1_000_000;
/// Generate a mapping to be used by `WeightedChoice`
fn gen_mapping() -> Vec<Weighted<usize>> {
DATA.iter()
.enumerate()
.map(|(i, &(_, p))| Weighted {
// `WeightedChoice` requires `u32` weights rather than raw probabilities. For each
// probability, we convert it to a `u32` weight, and associate it with an index. We
// multiply by a constant because small numbers such as 0.2 when casted to `u32`
// become `0`. This conversion decreases the accuracy of the mapping, which is why we
// provide an implementation which uses `f64`s for the best accuracy.
weight: (p * 1_000_000_000.0) as u32,
item: i,
})
.collect()
}
/// Generate a mapping of the raw probabilities
fn gen_mapping_float() -> Vec<f64> {
// This does the work of `WeightedChoice::new`, splitting a number into various ranges. The
// `item` of `Weighted` is represented here merely by the probability's position in the `Vec`.
let mut running_total = 0.0;
DATA.iter()
.map(|&(_, p)| {
running_total += p;
running_total
})
.collect()
}
/// An implementation of `WeightedChoice` which uses probabilities rather than weights. Refer to
/// the `WeightedChoice` source for serious usage.
struct WcFloat {
mapping: Vec<f64>,
}
impl WcFloat {
fn new(mapping: &[f64]) -> Self {
Self {
mapping: mapping.to_vec(),
}
}
// This is roughly the same logic as `WeightedChoice::ind_sample` (though is likely slower)
fn search(&self, sample_prob: f64) -> usize {
let idx = self.mapping
.binary_search_by(|p| p.partial_cmp(&sample_prob).unwrap());
match idx {
Ok(i) | Err(i) => i,
}
}
}
impl IndependentSample<usize> for WcFloat {
fn ind_sample<R: Rng>(&self, rng: &mut R) -> usize {
// Because we know the total is exactly 1.0, we can merely use a raw float value.
// Otherwise caching `Range::new(0.0, running_total)` and sampling with
// `range.ind_sample(&mut rng)` is recommended.
let sample_prob = rng.next_f64();
self.search(sample_prob)
}
}
impl Sample<usize> for WcFloat {
fn sample<R: Rng>(&mut self, rng: &mut R) -> usize {
self.ind_sample(rng)
}
}
fn take_samples<R: Rng, T>(rng: &mut R, wc: &T) -> [usize; 8]
where
T: IndependentSample<usize>,
{
let mut counts = [0; 8];
for _ in 0..SAMPLES {
let sample = wc.ind_sample(rng);
counts[sample] += 1;
}
counts
}
fn print_mapping(counts: &[usize]) {
println!("Item | Expected | Actual ");
println!("-------+----------+----------");
for (&(name, expected), &count) in DATA.iter().zip(counts.iter()) {
let real = count as f64 / SAMPLES as f64;
println!("{:6} | {:.6} | {:.6}", name, expected, real);
}
}
fn main() {
let mut rng = weak_rng();
println!(" ~~~ U32 METHOD ~~~");
let mut mapping = gen_mapping();
let wc = WeightedChoice::new(&mut mapping);
let counts = take_samples(&mut rng, &wc);
print_mapping(&counts);
println!();
println!(" ~~~ FLOAT METHOD ~~~");
// initialize the float version of `WeightedChoice`
let mapping = gen_mapping_float();
let wc = WcFloat::new(&mapping);
let counts = take_samples(&mut rng, &wc);
print_mapping(&counts);
}
- Output:
~~~ U32 METHOD ~~~ Item | Expected | Actual -------+----------+---------- aleph | 0.200000 | 0.200195 beth | 0.166667 | 0.166182 gimel | 0.142857 | 0.142502 daleth | 0.125000 | 0.125503 he | 0.111111 | 0.110820 waw | 0.100000 | 0.100166 zayin | 0.090909 | 0.090927 heth | 0.063456 | 0.063705 ~~~ FLOAT METHOD ~~~ Item | Expected | Actual -------+----------+---------- aleph | 0.200000 | 0.199984 beth | 0.166667 | 0.166634 gimel | 0.142857 | 0.143218 daleth | 0.125000 | 0.124956 he | 0.111111 | 0.111047 waw | 0.100000 | 0.099805 zayin | 0.090909 | 0.090513 heth | 0.063456 | 0.063843
Scala
This algorithm consists of a concise two-line tail-recursive loop (def weighted). The rest of the code is for API robustness, testing and display. weightedProb is for the task as stated (0 < p < 1), and weightedFreq is the equivalent based on integer frequencies (f >= 0).
object ProbabilisticChoice extends App {
import scala.collection.mutable.LinkedHashMap
def weightedProb[A](prob: LinkedHashMap[A,Double]): A = {
require(prob.forall{case (_, p) => p > 0 && p < 1})
assume(prob.values.sum == 1)
def weighted(todo: Iterator[(A,Double)], rand: Double, accum: Double = 0): A = todo.next match {
case (s, i) if rand < (accum + i) => s
case (_, i) => weighted(todo, rand, accum + i)
}
weighted(prob.toIterator, scala.util.Random.nextDouble)
}
def weightedFreq[A](freq: LinkedHashMap[A,Int]): A = {
require(freq.forall{case (_, f) => f >= 0})
require(freq.values.sum > 0)
def weighted(todo: Iterator[(A,Int)], rand: Int, accum: Int = 0): A = todo.next match {
case (s, i) if rand < (accum + i) => s
case (_, i) => weighted(todo, rand, accum + i)
}
weighted(freq.toIterator, scala.util.Random.nextInt(freq.values.sum))
}
// Tests:
val probabilities = LinkedHashMap(
'aleph -> 1.0/5,
'beth -> 1.0/6,
'gimel -> 1.0/7,
'daleth -> 1.0/8,
'he -> 1.0/9,
'waw -> 1.0/10,
'zayin -> 1.0/11,
'heth -> 1759.0/27720
)
val frequencies = LinkedHashMap(
'aleph -> 200,
'beth -> 167,
'gimel -> 143,
'daleth -> 125,
'he -> 111,
'waw -> 100,
'zayin -> 91,
'heth -> 63
)
def check[A](original: LinkedHashMap[A,Double], results: Seq[A]) {
val freq = results.groupBy(x => x).mapValues(_.size.toDouble/results.size)
original.foreach{case (k, v) =>
val a = v/original.values.sum
val b = freq(k)
val c = if (Math.abs(a - b) < 0.001) "ok" else "**"
println(f"$k%10s $a%.4f $b%.4f $c")
}
println(" "*10 + f" ${1}%.4f ${freq.values.sum}%.4f")
}
println("Checking weighted probabilities:")
check(probabilities, for (i <- 1 to 1000000) yield weightedProb(probabilities))
println
println("Checking weighted frequencies:")
check(frequencies.map{case (a, b) => a -> b.toDouble}, for (i <- 1 to 1000000) yield weightedFreq(frequencies))
}
- Output:
Checking weighted probabilities:
'aleph 0.2000 0.2001 ok
'beth 0.1667 0.1665 ok
'gimel 0.1429 0.1430 ok
'daleth 0.1250 0.1248 ok
'he 0.1111 0.1112 ok
'waw 0.1000 0.1000 ok
'zayin 0.0909 0.0911 ok
'heth 0.0635 0.0632 ok
1.0000 1.0000
Checking weighted frequencies:
'aleph 0.2000 0.2000 ok
'beth 0.1670 0.1672 ok
'gimel 0.1430 0.1432 ok
'daleth 0.1250 0.1243 ok
'he 0.1110 0.1105 ok
'waw 0.1000 0.1002 ok
'zayin 0.0910 0.0913 ok
'heth 0.0630 0.0632 ok
1.0000 1.0000
Scheme
Using guile scheme 2.0.11.
(use-modules (ice-9 format))
(define (random-choice probs)
(define choice (random 1.0))
(define (helper val prob-lis)
(let ((nval (- val (cadar prob-lis))))
(if
(< nval 0)
(caar prob-lis)
(helper nval (cdr prob-lis)))))
(helper choice probs))
(define (add-result result delta table)
(cond
((null? table) (list (list result delta)))
((eq? (caar table) result)
(cons (list result (+ (cadar table) delta)) (cdr table)))
(#t (cons (car table) (add-result result delta (cdr table))))))
(define (choices trials probs)
(define (helper trial-num freq-table)
(if
(= trial-num trials)
freq-table
(helper
(+ trial-num 1)
(add-result (random-choice probs) (/ 1 trials) freq-table))))
(helper 0 '()))
(define (format-results probs results)
(for-each
(lambda (x)
(format
#t
"~10a~10,5f~10,5f~%"
(car x)
(cadr x)
(cadr (assoc (car x) results))))
probs))
(define probs
'((aleph 1/5) (beth 1/6) (gimel 1/7) (daleth 1/8)
(he 1/9) (waw 1/10) (zayin 1/11) (heth 1759/27720)))
(format-results probs (choices 1000000 probs))
Example output:
aleph 0.20000 0.20051 beth 0.16667 0.16680 gimel 0.14286 0.14231 daleth 0.12500 0.12538 he 0.11111 0.11136 waw 0.10000 0.09955 zayin 0.09091 0.09096 heth 0.06346 0.06313
Seed7
To reduce the runtime this program should be compiled.
$ include "seed7_05.s7i";
include "float.s7i";
const type: letter is new enum
aleph, beth, gimel, daleth, he, waw, zayin, heth
end enum;
const func string: str (in letter: aLetter) is
return [] ("aleph", "beth", "gimel", "daleth", "he", "waw", "zayin", "heth") [succ(ord(aLetter))];
enable_output(letter);
const array [letter] integer: table is [letter] (
5544, 4620, 3960, 3465, 3080, 2772, 2520, 1759);
const func letter: randomLetter is func
result
var letter: resultLetter is aleph;
local
var integer: number is 0;
begin
number := rand(1, 27720);
while number > table[resultLetter] do
number -:= table[resultLetter];
incr(resultLetter);
end while;
end func;
const proc: main is func
local
var integer: count is 0;
var letter: aLetter is aleph;
var array [letter] integer: occurrence is letter times 0;
begin
for count range 1 to 1000000 do
aLetter := randomLetter;
incr(occurrence[aLetter]);
end for;
writeln("Name Count Ratio Expected");
for aLetter range letter.first to letter.last do
writeln(aLetter rpad 7 <& occurrence[aLetter] lpad 6 <&
flt(occurrence[aLetter]) / 10000.9 digits 4 lpad 8 <& "%" <&
100.0 * flt(table[aLetter]) / 27720.0 digits 4 lpad 8 <& "%");
end for;
end func;Outout:
Name Count Ratio Expected aleph 199788 19.9770% 20.0000% beth 166897 16.6882% 16.6667% gimel 143103 14.3090% 14.2857% daleth 125060 12.5049% 12.5000% he 110848 11.0838% 11.1111% waw 99550 9.9541% 10.0000% zayin 90918 9.0910% 9.0909% heth 63836 6.3830% 6.3456%
Sidef
define TRIALS = 1e4;
func prob_choice_picker(options) {
var n = 0;
var a = [];
options.each { |k,v|
n += v;
a << [n, k];
}
func {
var r = 1.rand;
a.first{|e| r <= e[0] }[1];
}
}
var ps = Hash(
aleph => 1/5,
beth => 1/6,
gimel => 1/7,
daleth => 1/8,
he => 1/9,
waw => 1/10,
zayin => 1/11
)
ps{:heth} = (1 - ps.values.sum)
var picker = prob_choice_picker(ps)
var results = Hash()
TRIALS.times {
results{picker()} := 0 ++;
}
say "Event Occurred Expected Difference";
for k,v in (results.sort_by {|k| results{k} }.reverse) {
printf("%-6s %f %f %f\n",
k, v/TRIALS, ps{k},
abs(v/TRIALS - ps{k})
);
}
- Output:
Event Occurred Expected Difference aleph 0.196300 0.200000 0.003700 beth 0.165600 0.166667 0.001067 gimel 0.143700 0.142857 0.000843 daleth 0.123900 0.125000 0.001100 he 0.111800 0.111111 0.000689 waw 0.101900 0.100000 0.001900 zayin 0.088100 0.090909 0.002809 heth 0.068800 0.063456 0.005344
SparForte
As a structured script.
#!/usr/local/bin/spar
pragma annotate( summary, "randdist" )
@( description, "Given a mapping between items and their required" )
@( description, "probability of occurrence, generate a million items" )
@( description, "randomly subject to the given probabilities and compare" )
@( description, "the target probability of occurrence versus the" )
@( description, "generated values." )
@( description, "" )
@( description, "The total of all the probabilities should equal one." )
@( description, "(Because floating point arithmetic is involved this is" )
@( description, "subject to rounding errors). Use the following mapping" )
@( description, "to test your programs: aleph 1/5.0, beth 1/6.0," )
@( description, "gimel 1/7.0, daleth 1/8.0, he 1/9.0, waw 1/10.0" )
@( description, "zayin 1/11.0, heth 1759/27720 adjusted so that" )
@( description, "probabilities add to 1" )
@( see_also, "http://rosettacode.org/wiki/Probabilistic_choice" )
@( author, "Ken O. Burtch" );
pragma license( unrestricted );
pragma restriction( no_external_commands );
procedure randdist is
trials : constant positive := 1_000_000;
type outcome is (aleph, beth, gimel, daleth, he, waw, zayin, heth);
pr : constant array(aleph..heth) of float :=
(1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1 );
samples : array(aleph..heth) of natural := (0, 0, 0, 0, 0, 0, 0, 0);
random_value : float;
begin
for try in 1..trials loop
random_value := numerics.random;
for i in arrays.first( pr )..arrays.last( pr ) loop
if random_value <= pr(i) then
samples(i) := samples(i) + 1;
exit;
else
random_value := @ - pr(i);
end if;
end loop;
end loop;
-- Show results
for i in arrays.first( pr )..arrays.last( pr ) loop
put( i ) @ ( " " ) @ ( float( samples( i ) ) / float( trials ) );
if i = heth then
put_line( " rest" );
else
put_line( pr(i) );
end if;
end loop;
end randdist;
- Output:
$ spar randdist aleph 2.00260000000000E-01 2.00000000000000E-01 beth 1.66376000000000E-01 1.66666666666667E-01 gimel 1.42698000000000E-01 1.42857142857143E-01 daleth 1.25408000000000E-01 1.25000000000000E-01 he 1.11311000000000E-01 1.11111111111111E-01 waw 9.97980000000000E-02 1.00000000000000E-01 zayin 9.09570000000000E-02 9.09090909090909E-02 heth 6.31920000000000E-02 rest
Stata
clear
mata
letters="aleph","beth","gimel","daleth","he","waw","zayin","heth"
a=letters[rdiscrete(10000,1,(1/5,1/6,1/7,1/8,1/9,1/10,1/11,1759/27720))]'
st_addobs(10000)
st_addvar("str10","a")
st_sstore(.,.,a)
end
Tcl
package require Tcl 8.5
set map [dict create]
set sum 0.0
foreach name {aleph beth gimel daleth he waw zayin} \
prob {1/5.0 1/6.0 1/7.0 1/8.0 1/9.0 1/10.0 1/11.0} \
{
set prob [expr $prob]
set sum [expr {$sum + $prob}]
dict set map $name [dict create probability $prob limit $sum count 0]
}
dict set map heth [dict create probability [expr {1.0 - $sum}] limit 1.0 count 0]
set samples 1000000
for {set i 0} {$i < $samples} {incr i} {
set n [expr {rand()}]
foreach name [dict keys $map] {
if {$n <= [dict get $map $name limit]} {
set count [dict get $map $name count]
dict set map $name count [incr count]
break
}
}
}
puts "using $samples samples:"
puts [format "%-10s %-21s %-9s %s" "" expected actual difference]
dict for {name submap} $map {
dict with submap {
set actual [expr {$count * 1.0 / $samples}]
puts [format "%-10s %-21s %-9s %4.2f%%" $name $probability $actual \
[expr {abs($actual - $probability)/$probability*100.0}]
]
}
}
using 1000000 samples:
expected actual difference
aleph 0.2 0.199641 0.18%
beth 0.16666666666666666 0.1674 0.44%
gimel 0.14285714285714285 0.143121 0.18%
daleth 0.125 0.124864 0.11%
he 0.1111111111111111 0.111036 0.07%
waw 0.1 0.100021 0.02%
zayin 0.09090909090909091 0.09018 0.80%
heth 0.06345598845598843 0.063737 0.44%
Ursala
The stochasm library function used here constructs a weighted non-deterministic choice of a set of functions. The pseudo-random number generator is a 64 bit Mersenne twistor implemented by the run time system.
#import std
#import nat
#import flo
outcomes = <'aleph ','beth ','gimel ','daleth','he ','waw ','zayin ','heth '>
probabilities = ^lrNCT(~&,minus/1.+ plus:-0) div/*1. float* skip/5 iota12
simulation =
^(~&rn,div+ float~~rmPlX)^*D/~& iota; ^A(~&h,length)*K2+ * stochasm@p/probabilities !* outcomes
format =
:/' frequency probability'+ * ^lrlrTPT/~&n (printf/'%12.8f')^~/~&m outcomes-$probabilities@n
#show+
results = format simulation 1000000output:
frequency probability daleth 0.12484500 0.12500000 beth 0.16680600 0.16666667 aleph 0.19973700 0.20000000 waw 0.10016900 0.10000000 gimel 0.14293100 0.14285714 he 0.11131100 0.11111111 zayin 0.09104700 0.09090909 heth 0.06315400 0.06345599
VBScript
Derived from the BBC BASIC version
item = Array("aleph","beth","gimel","daleth","he","waw","zayin","heth")
prob = Array(1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0, 1759/27720)
Dim cnt(7)
'Terminate script if sum of probabilities <> 1.
sum = 0
For i = 0 To UBound(prob)
sum = sum + prob(i)
Next
If sum <> 1 Then
WScript.Quit
End If
For trial = 1 To 1000000
r = Rnd(1)
p = 0
For i = 0 To UBound(prob)
p = p + prob(i)
If r < p Then
cnt(i) = cnt(i) + 1
Exit For
End If
Next
Next
WScript.StdOut.Write "item" & vbTab & "actual" & vbTab & vbTab & "theoretical"
WScript.StdOut.WriteLine
For i = 0 To UBound(item)
WScript.StdOut.Write item(i) & vbTab & FormatNumber(cnt(i)/1000000,6) & vbTab & FormatNumber(prob(i),6)
WScript.StdOut.WriteLine
Next
- Output:
item actual theoretical aleph 0.199755 0.200000 beth 0.166861 0.166667 gimel 0.143240 0.142857 daleth 0.124474 0.125000 he 0.110879 0.111111 waw 0.100341 0.100000 zayin 0.090745 0.090909 heth 0.063705 0.063456
V (Vlang)
import rand
const (
item_arr = ["aleph","beth","gimel","daleth","he","waw","zayin","heth"]
prob_arr = [1/5.0, 1/6.0, 1/7.0, 1/8.0, 1/9.0, 1/10.0, 1/11.0, 1759/27720.0]
)
fn main() {
mut cnt_arr := [8]f64{}
mut rnum, mut prob := f64(0), f64(0)
for _ in 1 .. 1000001 {
rnum = rand.f64n(10)! / 10
prob = 0
for idx in 0 .. prob_arr.len {
prob += prob_arr[idx]
if rnum < prob {
cnt_arr[idx]++
break
}
}
}
println("Item Expected Actual")
for idx in 0 .. item_arr.len {
println("${item_arr[idx]} ${prob_arr[idx]:.6f} ${cnt_arr[idx] / 1000000.0:.6f}")
}
}
- Output:
Item Expected Actual aleph 0.200000 0.199607 beth 0.166667 0.166164 gimel 0.142857 0.142826 daleth 0.125000 0.125384 he 0.111111 0.111604 waw 0.100000 0.100302 zayin 0.090909 0.090505 heth 0.063456 0.063608
Wren
import "random" for Random
import "./fmt" for Fmt
var letters = ["aleph", "beth", "gimel", "daleth", "he", "waw", "zayin", "heth"]
var actual = [0] * 8
var probs = [1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 0]
var cumProbs = [0] * 8
cumProbs[0] = probs[0]
for (i in 1..6) cumProbs[i] = cumProbs[i-1] + probs[i]
cumProbs[7] = 1
probs[7] = 1 - cumProbs[6]
var n = 1e6
var rand = Random.new()
(1..n).each { |i|
var r = rand.float()
var index = (r <= cumProbs[0]) ? 0 :
(r <= cumProbs[1]) ? 1 :
(r <= cumProbs[2]) ? 2 :
(r <= cumProbs[3]) ? 3 :
(r <= cumProbs[4]) ? 4 :
(r <= cumProbs[5]) ? 5 :
(r <= cumProbs[6]) ? 6 : 7
actual[index] = actual[index] + 1
}
var sumActual = 0
System.print("Letter\t Actual Expected")
System.print("------\t-------- --------")
for (i in 0..7) {
var generated = actual[i]/n
Fmt.print("$s\t$8.6f $8.6f", letters[i], generated, probs[i])
sumActual = sumActual + generated
}
System.print("\t-------- --------")
Fmt.print("\t$8.6f 1.000000", sumActual)
- Output:
Sample run:
Letter Actual Expected ------ -------- -------- aleph 0.200037 0.200000 beth 0.166643 0.166667 gimel 0.143012 0.142857 daleth 0.125219 0.125000 he 0.111183 0.111111 waw 0.099510 0.100000 zayin 0.091015 0.090909 heth 0.063381 0.063456 -------- -------- 1.000000 1.000000
XPL0
include c:\cxpl\codes;
def Size = 10_000_000;
int Tbl(12+1);
int I, J, N;
real X, S0, S1;
[for J:= 5 to 12 do Tbl(J):= 0;
for I:= 0 to 1_000_000-1 do \generate one million items
[N:= Ran(Size);
for J:= 5 to 11 do
[N:= N - Size/J;
if N < 0 then [Tbl(J):= Tbl(J)+1; J:= 100];
];
if J=12 then Tbl(12):= Tbl(12)+1;
];
S0:= 0.0; S1:= 0.0;
for J:= 5 to 11 do
[X:= 1.0/float(J); RlOut(0, X); S0:= S0+X;
X:= float(Tbl(J)) / 1_000_000.0; RlOut(0, X); S1:= S1+X;
CrLf(0);
];
X:= 1759.0 / 27720.0; RlOut(0, X); S0:= S0+X;
X:= float(Tbl(12)) / 1_000_000.0; RlOut(0, X); S1:= S1+X;
CrLf(0);
Text(0, " ------- -------
");
RlOut(0, S0); RlOut(0, S1);
]Output:
0.20000 0.20012
0.16667 0.16679
0.14286 0.14305
0.12500 0.12510
0.11111 0.11113
0.10000 0.09990
0.09091 0.09077
0.06346 0.06313
------- -------
1.00000 1.00000
Yabasic
dim letters$(7)
data "aleph", "beth", "gimel", "daleth", "he", "waw", "zayin", "heth"
letters$(0) = "aleph"
letters$(1) = "beth"
letters$(2) = "gimel"
letters$(3) = "daleth"
letters$(4) = "he"
letters$(5) = "waw"
letters$(6) = "zayin"
letters$(7) = "heth"
dim actual(7)
dim probs(7)
probs(0) = 1/5.0
probs(1) = 1/6.0
probs(2) = 1/7.0
probs(3) = 1/8.0
probs(4) = 1/9.0
probs(5) = 1/10.0
probs(6) = 1/11.0
probs(7) = 1759/27720
dim cumProbs(7)
cumProbs(0) = probs(0)
for i = 1 to 6
cumProbs(i) = cumProbs(i - 1) + probs(i)
next i
cumProbs(7) = 1.0
n = 1000000
for test = 1 to n
r = ran(1)
p = 0.0
for i = 1 to arraysize(probs(),1)
p = p + probs(i)
if r < p then
actual(i) = actual(i) + 1
break
end if
next i
next t
sumActual = 0.0
tab$ = chr$(9)
print "Letter Actual Expected"
print "------ -------- --------"
for i = 0 to 7
print letters$(i), tab$,
print actual(i)/n using "#.######",
sumActual = sumActual + actual(i)/n
print probs(i) using "#.######"
next i
print " -------- --------"
print " ", sumActual using "#.######", tab$, "1.000000"
endzkl
var names=T("aleph", "beth", "gimel", "daleth",
"he", "waw", "zayin", "heth");
var ptable=T(5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0).apply('/.fp(1.0));
ptable=ptable.append(1.0-ptable.sum(0.0)); // add last weight to sum to 1.0
var [const] N=ptable.len();
fcn ridx{ i:=0; s:=(0.0).random(1);
while((s-=ptable[i]) > 0) { i+=1 }
i
}
const M=0d1_000_000;
var r=(0).pump(N,List,T(Ref,0)); // list of references to int 0
(0).pump(M,Void,fcn{r[ridx()].inc()}); // 1,000,000 weighted random #s
r=r.apply("value").apply("toFloat"); // (reference to int)-->int-->float
println(" Name Count Ratio Expected");
foreach i in (N){
"%6s%7d %7.4f%% %7.4f%%".fmt(names[i], r[i], r[i]/M*100,
ptable[i]*100).println();
}- Output:
Name Count Ratio Expected
aleph 200214 20.0214% 20.0000%
beth 166399 16.6399% 16.6667%
gimel 143100 14.3100% 14.2857%
daleth 125197 12.5197% 12.5000%
he 111167 11.1167% 11.1111%
waw 100097 10.0097% 10.0000%
zayin 90692 9.0692% 9.0909%
heth 63162 6.3162% 6.3456%
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