Unbias a random generator
You are encouraged to solve this task according to the task description, using any language you may know.
Given a weighted one bit generator of random numbers where the probability of a one occuring, , is not the same as , the probability of a zero occuring, the probability of the occurrence of a one followed by a zero is . This is the same as the probability of a zero followed by a one: .
Task Details
- Use your language's random number generator to create a function/method/subroutine/... randN that returns a one or a zero, but with one occurring, on average, 1 out of N times, where N is an integer from the range 3 to 6 inclusive.
- Create a function unbiased that uses only randN as its source of randomness to become an unbiased generator of random ones and zeroes.
- For N over its range, generate and show counts of the outputs of randN and unbiased(randN).
The actual unbiasing should be done by generating two numbers at a time from randN and only returning a 1 or 0 if they are different. As long as you always return the first number or always return the second number, the probabilities discussed above should take over the biased probability of randN.
D
<lang d>import std.stdio, std.random;
bool biased(int n) {
return uniform(0.0, 1.0) < (1.0 / n);
}
bool unbiased(int n) {
bool a, b;
do {
a = biased(n);
b = biased(n);
} while (a == b);
return a;
}
void main() {
int M = 50_000;
foreach (n; 3 .. 7) {
int c1, c2;
foreach (i; 0 .. M) {
c1 += biased(n);
c2 += unbiased(n);
}
writefln("%d: %2.2f%% %2.2f%%", n, 100.0*c1/M, 100.0*c2/M);
}
}</lang> Output:
3: 33.12% 50.15% 4: 25.13% 50.02% 5: 19.61% 50.01% 6: 16.70% 49.98%
J
<lang j>randN=: 0 = ? unbiased=: i.@# { ::$: 2 | 0 3 -.~ _2 #.\ 4&* randN@# ]</lang>
Example use:
<lang j> randN 10#6 1 0 0 0 1 0 0 0 0 0
unbiased 10#6
1 0 0 1 0 0 1 0 1 1</lang>
Some example counts (these are counts of the number of 1s which appear in a test involving 100 random numbers):
<lang j> +/randN 100#3 30
+/randN 100#4
20
+/randN 100#5
18
+/randN 100#6
18
+/unbiased 100#3
49
+/unbiased 100#4
46
+/unbiased 100#5
49
+/unbiased 100#6
47</lang>
Note that these results are random. For example, a re-run of +/randN 100#5 gave 25 as its result, and a re-run of +/unbiased 100#5 gave 52 as its result.
Java
<lang java>public class Bias {
public static boolean biased(int n) {
return Math.random() < 1.0 / n;
}
public static boolean unbiased(int n) {
boolean a, b;
do {
a = biased(n);
b = biased(n);
} while (a == b);
return a;
}
public static void main(String[] args) {
final int M = 50000;
for (int n = 3; n < 7; n++) {
int c1 = 0, c2 = 0;
for (int i = 0; i < M; i++) {
c1 += biased(n) ? 1 : 0;
c2 += unbiased(n) ? 1 : 0;
}
System.out.format("%d: %2.2f%% %2.2f%%\n",
n, 100.0*c1/M, 100.0*c2/M);
}
}
}</lang> Output:
3: 33,11% 50,23% 4: 24.97% 49.78% 5: 20.05% 50.00% 6: 17.00% 49.88%
PicoLisp
<lang PicoLisp>(de randN (N)
(if (= 1 (rand 1 N)) 1 0) )
(de unbiased (N)
(use (A B)
(while
(=
(setq A (randN N))
(setq B (randN N)) ) )
A ) )</lang>
Test: <lang PicoLisp>(for N (range 3 6)
(tab (2 1 7 2 7 2)
N ":"
(format
(let S 0 (do 10000 (inc 'S (randN N))))
2 )
"%"
(format
(let S 0 (do 10000 (inc 'S (unbiased N))))
2 )
"%" ) )</lang>
Output:
3: 33.21 % 50.48 % 4: 25.06 % 49.79 % 5: 20.04 % 49.75 % 6: 16.32 % 49.02 %
Python
<lang python>from __future__ import print_function import random
def randN(N):
" 1,0 random generator factory with 1 appearing 1/N'th of the time"
choose = [1] + [0] * (N-1)
def rand():
return random.choice( rand.choose )
rand.choose = choose
return rand
def unbiased(biased):
'uses a biased() generator of 1 or 0, to create an unbiased one'
this, that = biased(), biased()
while this == that: # Loop until 10 or 01
this, that = biased(), biased()
return this # return the first
if __name__ == '__main__':
from collections import namedtuple
Stats = namedtuple('Stats', 'count1 count0 percent')
for N in range(3, 7):
biased = randN(N)
v = [biased() for x in range(1000000)]
v1, v0 = v.count(1), v.count(0)
print ( "Biased(%i) = %r" % (N, Stats(v1, v0, 100. * v1/(v1 + v0))) )
v = [unbiased(biased) for x in range(1000000)]
v1, v0 = v.count(1), v.count(0)
print ( " Unbiased = %r" % (Stats(v1, v0, 100. * v1/(v1 + v0)), ) )</lang>
Sample output
Biased(3) = Stats(count1=331800, count0=668200, percent=33.18) Unbiased = Stats(count1=499740, count0=500260, percent=49.973999999999997) Biased(4) = Stats(count1=249770, count0=750230, percent=24.977) Unbiased = Stats(count1=499707, count0=500293, percent=49.970700000000001) Biased(5) = Stats(count1=199764, count0=800236, percent=19.976400000000002) Unbiased = Stats(count1=499456, count0=500544, percent=49.945599999999999) Biased(6) = Stats(count1=167561, count0=832439, percent=16.7561) Unbiased = Stats(count1=499963, count0=500037, percent=49.996299999999998)
Tcl
<lang tcl># 1,0 random generator factory with 1 appearing 1/N'th of the time proc randN n {expr {rand()*$n < 1}}
- uses a biased generator of 1 or 0, to create an unbiased one
proc unbiased {biased} {
while 1 {
if {[set a [eval $biased]] != [eval $biased]} {return $a}
}
}
for {set n 3} {$n <= 6} {incr n} {
set biased [list randN $n]
for {set i 0;array set c {0 0 1 0}} {$i < 1000000} {incr i} {
incr c([eval $biased])
} puts [format " biased %d => #0=%d #1=%d ratio=%.2f%%" $n $c(0) $c(1) \
[expr {100.*$c(1)/$i}]]
for {set i 0;array set c {0 0 1 0}} {$i < 1000000} {incr i} {
incr c([unbiased $biased])
} puts [format "unbiased %d => #0=%d #1=%d ratio=%.2f%%" $n $c(0) $c(1) \
[expr {100.*$c(1)/$i}]] }</lang> Sample output:
biased 3 => #0=667076 #1=332924 ratio=33.29% unbiased 3 => #0=500263 #1=499737 ratio=49.97% biased 4 => #0=750470 #1=249530 ratio=24.95% unbiased 4 => #0=500644 #1=499356 ratio=49.94% biased 5 => #0=800243 #1=199757 ratio=19.98% unbiased 5 => #0=500878 #1=499122 ratio=49.91% biased 6 => #0=833623 #1=166377 ratio=16.64% unbiased 6 => #0=500518 #1=499482 ratio=49.95%