# Sleeping Beauty problem

**Sleeping Beauty problem**

You are encouraged to solve this task according to the task description, using any language you may know.

- Background on the task

In decision theory, The Sleeping Beauty Problem
is a problem invented by Arnold Zoboff and first publicized on Usenet. The experimental
subject, named Sleeping Beauty, agrees to an experiment as follows:
Sleeping Beauty volunteers to be put into a deep sleep on a Sunday. There is then a fair coin toss.
If this coin toss comes up heads, Sleeping Beauty wakes once (on Monday) and is asked to
estimate the probability that the coin toss was heads. Her estimate is recorded and she is
then put back to sleep for 2 days until Wednesday, at which time the experiment's results are tallied.

If instead the coin toss is tails, Sleeping Beauty wakes as before on Monday and asked to
estimate the probability the coin toss was heads, but is then given a drug which makes her forget
that she had been woken on Monday before being put back to sleep again. She then wakes only 1 day
later, on Tuesday. She is then asked (on Tuesday) again to guess the probability that the coin toss
was heads or tails. She is then put back to sleep and awakes as before 1 day later, on Wednesday.

Some decision makers have argued that since the coin toss was fair Sleeping Beauty should always estimate the probability of heads as 1/2, since she does not have any additional information. Others have disagreed, saying that if Sleeping Beauty knows the study design she also knows that she is twice as likely to wake up and be asked to estimate the coin flip on tails than on heads, so the estimate should be 1/3 heads.

- Task

Given the above problem, create a Monte Carlo estimate of the actual results. The program should find the proportion of heads on waking and asking Sleeping Beauty for an estimate, as a credence or as a percentage of the times Sleeping Beauty is asked the question.

## Contents

## 11l[edit]

F sleeping_beauty_experiment(repetitions)

‘

Run the Sleeping Beauty Problem experiment `repetitions` times, checking to see

how often we had heads on waking Sleeping Beauty.

’

V gotheadsonwaking = 0

V wakenings = 0

L 0 .< repetitions

V coin_result = random:choice([‘heads’, ‘tails’])

// On Monday, we check if we got heads.

wakenings++

I coin_result == ‘heads’

gotheadsonwaking++

// If tails, we do this again, but of course we will not add as if it was heads..

I coin_result == ‘tails’

wakenings++

I coin_result == ‘heads’

gotheadsonwaking++ // never done

print(‘Wakenings over ’repetitions‘ experiments: ’wakenings)

R Float(gotheadsonwaking) / wakenings

V CREDENCE = sleeping_beauty_experiment(1'000'000)

print(‘Results of experiment: Sleeping Beauty should estimate a credence of: ’CREDENCE)

- Output:

Wakenings over 1000000 experiments: 1500892 Results of experiment: Sleeping Beauty should estimate a credence of: 0.332540916

## Arturo[edit]

sleepingBeauty: function [reps][

wakings: 0

heads: 0

do.times: reps [

coin: random 0 1

wakings: wakings + 1

if? coin = 0 -> heads: heads + 1

else -> wakings: wakings + 1

]

print ["Wakings over" reps "repetitions =" wakings]

return 100.0 * heads//wakings

]

pc: sleepingBeauty 100000

print ["Percentage probability of heads on waking =" pc "%"]

- Output:

Wakings over 100000 repetitions = 150096 Percentage probability of heads on waking = 33.24805457840316 %

## BASIC[edit]

### BASIC256[edit]

iteraciones = 1000000

cara = 0

dormir = 0

for i = 1 to iteraciones

lanza_moneda = int(rand * 2)

dormir = dormir + 1

if lanza_moneda = 1 then

cara = cara + 1

else

dormir = dormir + 1

end if

next i

print "Wakings over "; iteraciones; " repetitions = "; dormir

print "Percentage probability of heads on waking = "; (cara/dormir*100); "%"

end

- Output:

Igual que la entrada de FreeBASIC.

### FreeBASIC[edit]

Const iteraciones = 1000000

Randomize Timer

Dim As Uinteger cara = 0, dormir = 0

For i As Uinteger = 1 To iteraciones

Dim As integer lanza_moneda = Int(Rnd * 2) + 1

dormir += 1

if lanza_moneda = 1 then cara += 1 else dormir += 1

Next i

Print Using "Wakings over #####,### repetitions = #####,###"; iteraciones ; dormir

Print using "Percentage probability of heads on waking = ###.######%"; (cara/dormir*100)'; "%"

Sleep

- Output:

Wakings over 1,000,000 repetitions = 1,499,718 Percentage probability of heads on waking = 33.358405%

### GW-BASIC[edit]

In this simulation, Sleeping Beauty flips a coin of her own.

10 RANDOMIZE TIMER

20 MONDAY = 0 : TUESDAY = 1

30 HEADS = 0 : TAILS = 1

40 FOR SB = 1 TO 300000!

50 IF COIN = HEADS THEN GOSUB 150 ELSE GOSUB 210

60 COIN = INT(RND*2)

70 NEXT SB

80 PRINT "Sleeping Beauty was put through this experiment ";SB-1;" times."

90 PRINT "She was awoken ";AWAKE;" times."

100 PRINT "She guessed heads ";CHEADS+WHEADS;" times."

110 PRINT "Those guesses were correct ";CHEADS;" times. ";100*CHEADS/(CHEADS+WHEADS);"%"

120 PRINT "She guessed tails ";WTAILS+CTAILS;" times."

130 PRINT "Those guesses were correct ";CTAILS;" times. ";100*CTAILS/(CTAILS+WTAILS);"%"

140 END

150 REM interview if the coin came up heads

160 AWAKE = AWAKE + 1

170 NHEADS = NHEADS + 1

180 GUESS = INT(RND*2)

190 IF GUESS = HEADS THEN CHEADS = CHEADS + 1 ELSE WTAILS = WTAILS + 1

200 RETURN

210 REM interviews if the coin came up tails

220 FOR DAY = MONDAY TO TUESDAY

230 AWAKE = AWAKE + 1

240 GUESS = INT(RND*2)

250 IF GUESS = HEADS THEN WHEADS = WHEADS + 1 ELSE CTAILS = CTAILS + 1

260 NEXT DAY

270 RETURN

- Output:

Sleeping Beauty was put through this experiment 300000 times. She was awoken 449981 times. She guessed heads 224569 times. Those guesses were correct 74985 times. 33.39063 % She guessed tails 225412 times. Those guesses were correct 150378 times. 66.71251 %

### Yabasic[edit]

iteraciones = 1000000

cara = 0

dormir = 0

for i = 1 to iteraciones

lanza_moneda = int(ran(2))

dormir = dormir + 1

if lanza_moneda = 1 then cara = cara + 1 else dormir = dormir + 1 endif

next i

print "Wakings over ", iteraciones, " repetitions = ", dormir

print "Percentage probability of heads on waking = ", (cara/dormir*100), "%"

end

- Output:

Igual que la entrada de FreeBASIC.

## C++[edit]

#include <iostream>

#include <random>

int main() {

std::cout.imbue(std::locale(""));

const int experiments = 1000000;

std::random_device dev;

std::default_random_engine engine(dev());

std::uniform_int_distribution<int> distribution(0, 1);

int heads = 0, wakenings = 0;

for (int i = 0; i < experiments; ++i) {

++wakenings;

switch (distribution(engine)) {

case 0: // heads

++heads;

break;

case 1: // tails

++wakenings;

break;

}

}

std::cout << "Wakenings over " << experiments

<< " experiments: " << wakenings << '\n';

std::cout << "Sleeping Beauty should estimate a credence of: "

<< double(heads) / wakenings << '\n';

}

- Output:

Wakenings over 1,000,000 experiments: 1,500,090 Sleeping Beauty should estimate a credence of: 0.333253

## CLU[edit]

% This program needs to be merged with PCLU's "misc" library

% to use the random number generator.

experiment = cluster is run

rep = null

own awake: int := 0

own awake_heads: int := 0

% Returns true if heads, false if tails

coin_toss = proc () returns (bool)

return(random$next(2)=1)

end coin_toss

% Do the experiment once

do_experiment = proc ()

heads: bool := coin_toss()

% monday - wake up

awake := awake + 1

if heads then

awake_heads := awake_heads + 1

return

end

% tuesday - wake up if tails

awake := awake + 1

end do_experiment

% Run the experiment N times

run = proc (n: int) returns (real)

awake := 0

awake_heads := 0

for i: int in int$from_to(1,n) do

do_experiment()

end

return(real$i2r(awake_heads) / real$i2r(awake))

end run

end experiment

start_up = proc ()

N = 1000000

po: stream := stream$primary_output()

stream$puts(po, "Chance of waking up with heads: ")

chance: real := experiment$run(N)

stream$putl(po, f_form(chance, 1, 6))

end start_up

- Output:

Chance of waking up with heads: 0.333758

## Dyalect[edit]

let experiments = 10000

var heads = 0

var wakenings = 0

for _ in 1..experiments {

wakenings += 1

match rnd(min: 0, max: 10) {

<5 => heads += 1,

_ => wakenings += 1

}

}

print("Wakenings over \(experiments) experiments: \(wakenings)")

print("Sleeping Beauty should estimate a credence of: \(Float(heads) / Float(wakenings))")

## Excel[edit]

### LAMBDA[edit]

Binding the name SLEEPINGB to the lambda expression below in the Excel Workbook Name Manager:

(See LAMBDA: The ultimate Excel worksheet function)

SLEEPINGB

=LAMBDA(n,

LET(

headsWakes, LAMBDA(x,

IF(1 = x,

{1,1},

{0,2}

)

)(

RANDARRAY(n, 1, 0, 1, TRUE)

),

CHOOSE(

{1,2},

SUM(INDEX(headsWakes, 0, 1)),

SUM(INDEX(headsWakes, 0, 2))

)

)

)

- Output:

The pair of values in cells B2 and C2 both result from the application of SLEEPINGB in B2.

The credence value is returned as a ratio by the expression B2/C2 in cell D2,

with the format setting *Number > Fraction > Up to three digits*.

fx | =SLEEPINGB(1000000) | ||||
---|---|---|---|---|---|

A | B | C | D | ||

1 | Heads | Wakenings | Credence | ||

2 | Results | 500111 | 1499889 | 1/3 |

## F#[edit]

// Sleeping Beauty: Nigel Galloway. May 16th., 2021

let heads,woken=let n=System.Random() in {1..1000}|>Seq.fold(fun(h,w) g->match n.Next(2) with 0->(h+1,w+1) |_->(h,w+2))(0,0)

printfn "During 1000 tosses Sleeping Beauty woke %d times, %d times the toss was heads. %.0f%% of times heads had been tossed when she awoke" woken heads (100.0*float(heads)/float(woken))

- Output:

During 1000 tosses Sleeping Beauty woke 1519 times, 481 times the toss was heads. 32% of times heads had been tossed when she awoke

## Factor[edit]

USING: combinators.random io kernel math prettyprint ;

: sleeping ( n -- heads wakenings )

0 0 rot [ 1 + .5 [ [ 1 + ] dip ] [ 1 + ] ifp ] times ;

"Wakenings over 1,000,000 experiments: " write

1e6 sleeping dup . /f

"Sleeping Beauty should estimate a credence of: " write .

- Output:

Wakenings over 1,000,000 experiments: 1500127 Sleeping Beauty should estimate a credence of: 0.3332204540015612

## Go[edit]

package main

import (

"fmt"

"math/rand"

"rcu"

"time"

)

func sleepingBeauty(reps int) float64 {

wakings := 0

heads := 0

for i := 0; i < reps; i++ {

coin := rand.Intn(2) // heads = 0, tails = 1 say

wakings++

if coin == 0 {

heads++

} else {

wakings++

}

}

fmt.Printf("Wakings over %s repetitions = %s\n", rcu.Commatize(reps), rcu.Commatize(wakings))

return float64(heads) / float64(wakings) * 100

}

func main() {

rand.Seed(time.Now().UnixNano())

pc := sleepingBeauty(1e6)

fmt.Printf("Percentage probability of heads on waking = %f%%\n", pc)

}

- Output:

Sample run:

Wakings over 1,000,000 repetitions = 1,500,256 Percentage probability of heads on waking = 33.310582%

## Haskell[edit]

import Data.Monoid (Sum(..))

import System.Random (randomIO)

import Control.Monad (replicateM)

import Data.Bool (bool)

data Toss = Heads | Tails deriving Show

anExperiment toss =

moreWakenings <>

case toss of

Heads -> headsOnWaking

Tails -> moreWakenings

where

moreWakenings = (1,0)

headsOnWaking = (0,1)

main = do

tosses <- map (bool Heads Tails) <$> replicateM 1000000 randomIO

let (Sum w, Sum h) = foldMap anExperiment tosses

let ratio = fromIntegral h / fromIntegral w

putStrLn $ "Ratio: " ++ show ratio

*Main> main Ratio: 0.33339378051805013

## Julia[edit]

"""

Run the Sleeping Beauty Problem experiment `repetitions` times, checking to see

how often we had heads on waking Sleeping Beauty.

"""

function sleeping_beauty_experiment(repetitions)

gotheadsonwaking = 0

wakenings = 0

for _ in 1:repetitions

coin_result = rand(["heads", "tails"])

# On Monday, we check if we got heads.

wakenings += 1

if coin_result == "heads"

gotheadsonwaking += 1

end

# If tails, we do this again, but of course we will not add as if it was heads.

if coin_result == "tails"

wakenings += 1

if coin_result == "heads"

gotheadsonwaking += 1 # never done

end

end

end

# Show the number of times she was wakened.

println("Wakenings over ", repetitions, " experiments: ", wakenings)

# Return the number of correct bets SB made out of the total number

# of times she is awoken over all the experiments with that bet.

return gotheadsonwaking / wakenings

end

CREDENCE = sleeping_beauty_experiment(1_000_000)

println("Results of experiment: Sleeping Beauty should estimate a credence of: ", CREDENCE)

- Output:

Wakenings over 1000000 experiments: 1499534 Results of experiment: Sleeping Beauty should estimate a credence of: 0.33374768428058316

## Mathematica/Wolfram Language[edit]

ClearAll[SleepingBeautyExperiment]

SleepingBeautyExperiment[reps_Integer] := Module[{gotheadsonwaking, wakenings, coinresult},

gotheadsonwaking = 0;

wakenings = 0;

Do[

coinresult = RandomChoice[{"heads", "tails"}];

wakenings++;

If[coinresult === "heads",

gotheadsonwaking++;

,

wakenings++;

]

,

{reps}

];

Print["Wakenings over ", reps, " experiments: ", wakenings];

gotheadsonwaking/wakenings

]

out = [email protected][10^6];

Print["Results of experiment: Sleeping Beauty should estimate a credence of: ", out]

- Output:

Wakenings over 1000000 experiments: 1499714 Results of experiment: Sleeping Beauty should estimate a credence of: 0.333588

## Nim[edit]

import random

const N = 1_000_000

type Side {.pure.} = enum Heads, Tails

const Sides = [Heads, Tails]

randomize()

var onHeads, wakenings = 0

for _ in 1..N:

let side = sample(Sides)

inc wakenings

if side == Heads:

inc onHeads

else:

inc wakenings

echo "Wakenings over ", N, " experiments: ", wakenings

echo "Sleeping Beauty should estimate a credence of: ", onHeads / wakenings

- Output:

Wakenings over 1000000 experiments: 1499971 Sleeping Beauty should estimate a credence of: 0.3333591116094911

## Pascal[edit]

program sleepBeau;

uses

sysutils; //Format

const

iterations = 1000*1000;

fmt = 'Wakings over %d repetitions = %d'+#13#10+

'Percentage probability of heads on waking = %8.5f%%';

var

i,

heads,

wakings,

flip: Uint32;

begin

randomize;

for i :=1 to iterations do

Begin

flip := random(2)+1;//-- 1==heads, 2==tails

inc(wakings,1 + Ord(flip=2));

inc(heads,Ord(flip=1));

end;

writeln(Format(fmt,[iterations,wakings,heads/wakings*100]));

end.

- Output:

Wakings over 1000000 repetitions = 1499741 Percentage probability of heads on waking = 33.35636%

## Perl[edit]

use strict;

use warnings;

sub sleeping_beauty {

my($trials) = @_;

my($gotheadsonwaking,$wakenings);

$wakenings++ and rand > .5 ? $gotheadsonwaking++ : $wakenings++ for 1..$trials;

$wakenings, $gotheadsonwaking/$wakenings

}

my $trials = 1_000_000;

printf "Wakenings over $trials experiments: %d\nSleeping Beauty should estimate a credence of: %.4f\n", sleeping_beauty($trials);

- Output:

Wakenings over 1000000 experiments: 1499816 Sleeping Beauty should estimate a credence of: 0.333

## Phix[edit]

constant iterations = 1_000_000, fmt = """ Wakings over %,d repetitions = %,d Percentage probability of heads on waking = %f%% """ integer heads = 0, wakings = 0 for i=1 to iterations do integer flip = rand(2) -- 1==heads, 2==tails wakings += 1 + (flip==2) heads += (flip==1) end for printf(1,fmt,{iterations,wakings,heads/wakings*100})

- Output:

(You'll get the exact result less than 1% of the time!!)

Wakings over 1,000,000 repetitions = 1,500,000 Percentage probability of heads on waking = 33.333333%

## Python[edit]

### Procedural[edit]

from random import choice

def sleeping_beauty_experiment(repetitions):

"""

Run the Sleeping Beauty Problem experiment `repetitions` times, checking to see

how often we had heads on waking Sleeping Beauty.

"""

gotheadsonwaking = 0

wakenings = 0

for _ in range(repetitions):

coin_result = choice(["heads", "tails"])

# On Monday, we check if we got heads.

wakenings += 1

if coin_result == "heads":

gotheadsonwaking += 1

# If tails, we do this again, but of course we will not add as if it was heads..

if coin_result == "tails":

wakenings += 1

if coin_result == "heads":

gotheadsonwaking += 1 # never done

# Show the number of times she was wakened.

print("Wakenings over", repetitions, "experiments:", wakenings)

# Return the number of correct bets SB made out of the total number

# of times she is awoken over all the experiments with that bet.

return gotheadsonwaking / wakenings

CREDENCE = sleeping_beauty_experiment(1_000_000)

print("Results of experiment: Sleeping Beauty should estimate a credence of:", CREDENCE)

- Output:

Wakenings over 1000000 experiments: 1499765 Results of experiment: Sleeping Beauty should estimate a credence of: 0.333542254953276

### Functional[edit]

'''Sleeping Beauty Problem'''

from random import choice

from itertools import repeat

from functools import reduce

# experiment :: (Int, Int) -> IO (Int, Int)

def experiment(headsWakings):

'''A pair of counts updated by a coin flip.

'''

heads, wakings = headsWakings

return (

1 + heads, 1 + wakings

) if "h" == choice(["h", "t"]) else (

heads, 2 + wakings

)

# ------------------------- TEST -------------------------

# main :: IO ()

def main():

'''Observed results from one million runs.'''

n = 1_000_000

heads, wakes = applyN(n)(

experiment

)(

(0, 0)

)

print(

f'{wakes} wakenings over {n} experiments.\n'

)

print('Sleeping Beauty should estimate credence')

print(f'at around {round(heads/wakes, 3)}')

# ----------------------- GENERIC ------------------------

# applyN :: Int -> (a -> a) -> a -> a

def applyN(n):

'''n applications of f.

(Church numeral n).

'''

def go(f):

def ga(a, g):

return g(a)

def fn(x):

return reduce(ga, repeat(f, n), x)

return fn

return go

# MAIN ---

if __name__ == '__main__':

main()

- Output:

1500188 wakenings over 1000000 experiments. Sleeping Beauty should estimate credence at around 0.333

## Quackery[edit]

[ $ "bigrat.qky" loadfile ] now!

[ say "Number of trials: "

dup echo cr

0 ( heads count )

0 ( sleeps count )

rot times

[ 1+

2 random if

[ 1+ dip 1+ ] ]

say "Data: heads count: "

over echo cr

say " sleeps count: "

dup echo cr

say "Credence of heads: "

2dup 20 point$ echo$ cr

say " or approximately: "

10 round vulgar$ echo$ cr ] is trials ( n --> n/d )

1000000 trials

- Output:

Number of trials: 1000000 Data: heads count: 500212 sleeps count: 1500212 Credence of heads: 0.33342754224069664821 or approximately: 1/3

## R[edit]

There's nothing complicated here. Pretty much every language that resembles C is going to use virtually the same solution.

beautyProblem <- function(n)

{

wakeCount <- headCount <- 0

for(i in seq_len(n))

{

wakeCount <- wakeCount + 1

if(sample(c("H", "T"), 1) == "H") headCount <- headCount + 1 else wakeCount <- wakeCount + 1

}

headCount/wakeCount

}

print(beautyProblem(10000000))

- Output:

[1] 0.3335838

## Raku[edit]

sub sleeping-beauty ($trials) {

my $gotheadsonwaking = 0;

my $wakenings = 0;

^$trials .map: {

given <Heads Tails>.roll {

++$wakenings;

when 'Heads' { ++$gotheadsonwaking }

when 'Tails' { ++$wakenings }

}

}

say "Wakenings over $trials experiments: ", $wakenings;

$gotheadsonwaking / $wakenings

}

say "Results of experiment: Sleeping Beauty should estimate a credence of: ", sleeping-beauty(1_000_000);

- Output:

Wakenings over 1000000 experiments: 1500040 Results of experiment: Sleeping Beauty should estimate a credence of: 0.333298

## Red[edit]

Red ["Sleeping Beauty problem"]

experiments: 1'000'000

heads: awakenings: 0

loop experiments [

awakenings: awakenings + 1

either 1 = random 2 [heads: heads + 1] [awakenings: awakenings + 1]

]

print ["Awakenings over" experiments "experiments:" awakenings]

print ["Probability of heads on waking:" heads / awakenings]

- Output:

Awakenings over 1000000 experiments: 1500063 Probability of heads on waking: 0.3332773356852345

## REXX[edit]

When using Regina REXX, the seed specified (for **random**) was **46**.

/*REXX pgm uses a Monte Carlo estimate for the results for the Sleeping Beauty problem. */

parse arg n seed . /*obtain optional arguments from the CL*/

if n=='' | n=="," then n= 1000000 /*Not specified? Then use the default.*/

if datatype(seed, 'W') then call random ,,seed /* Specified? Then use as RAND seed*/

awake= 0 /* " " " " awakened. */

do #=0 for n /*perform experiment: 1 million times?*/

if random(,1) then awake= awake + 1 /*Sleeping Beauty is awoken. */

else #= # + 1 /* " " keeps sleeping. */

end /*#*/ /* [↑] RANDOM returns: 0 or 1 */

say 'Wakenings over ' commas(n) " repetitions: " commas(#)

say 'The percentage probability of heads on awakening: ' (awake / # * 100)"%"

exit 0 /*stick a fork in it, we're all done. */

/*──────────────────────────────────────────────────────────────────────────────────────*/

commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ?

- output when using the input of:
`, 46`

Wakenings over 1,000,000 repetitions: 1,500,000 The percentage probability of heads on awakening: 33.3333333%

## Ruby[edit]

def sleeping_beauty_experiment(n)

coin = [:heads, :tails]

gotheadsonwaking = 0

wakenings = 0

n.times do

wakenings += 1

coin.sample == :heads ? gotheadsonwaking += 1 : wakenings += 1

end

puts "Wakenings over #{n} experiments: #{wakenings}"

gotheadsonwaking / wakenings.to_f

end

puts "Results of experiment: Sleeping Beauty should estimate

a credence of: #{sleeping_beauty_experiment(1_000_000)}"

- Output:

Wakenings over 1000000 experiments: 1499604 Results of experiment: Sleeping Beauty should estimate a credence of: 0.3336854262858728

## Swift[edit]

let experiments = 1000000

var heads = 0

var wakenings = 0

for _ in (1...experiments) {

wakenings += 1

switch (Int.random(in: 0...1)) {

case 0:

heads += 1

default:

wakenings += 1

}

}

print("Wakenings over \(experiments) experiments: \(wakenings)")

print("Sleeping Beauty should estimate a credence of: \(Double(heads) / Double(wakenings))")

- Output:

Wakenings over 1000000 experiments: 1500036 Sleeping Beauty should estimate a credence of: 0.3333013341013149

## Wren[edit]

import "random" for Random

import "/fmt" for Fmt

var rand = Random.new()

var sleepingBeauty = Fn.new { |reps|

var wakings = 0

var heads = 0

for (i in 0...reps) {

var coin = rand.int(2) // heads = 0, tails = 1 say

wakings = wakings + 1

if (coin == 0) {

heads = heads + 1

} else {

wakings = wakings + 1

}

}

Fmt.print("Wakings over $,d repetitions = $,d", reps, wakings)

return heads/wakings * 100

}

var pc = sleepingBeauty.call(1e6)

Fmt.print("Percentage probability of heads on waking = $f\%", pc)

- Output:

Sample run:

Wakings over 1,000,000 repetitions = 1,500,321 Percentage probability of heads on waking = 33.304806%