Your Choice Kills You

If you choose blue, you either need exactly $0$ out of the $4$ prisoners to choose blue or exactly $1$ out of the $4$ prisoners to choose blue to be in the minority and be able to leave jail.

The probability of exactly $0$ out of the $4$ prisoners choosing blue is $\frac{4 \choose 0}{2^4} = \frac{1}{16}$ , and the probability of exactly $1$ out of the $4$ prisoners choosing blue is $\frac{4 \choose 1}{2^4} = \frac{1}{4}$ .

The probability of either of these happening is $\frac{1}{16} + \frac{1}{4} = \frac{5}{16} = \boxed{31.25 \%}$ .

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Since the numbers are somewhat small, all the possible ways of the prisoner choices can also be drawn out. (The first ball is your choice, and the next $4$ balls are the other prisoner choices. The boxed choices are the situations where blue is in the minority and you will be able to leave jail.)

There are $5$ out of $16$ situations in which you would be able to leave jail, for a probability of $\frac{5}{16} = \boxed{31.25 \%}$

<?php
//treat 0 as blue, 1 as red
//each entry in the array represents the choices of the other 4 prisoners (0000 means they all chose blue, etc.)
$arr = array(0,1);
do{
$newarr = array();
foreach($arr as $a){
$newarr[] = $a.'0';
$newarr[] = $a.'1';
}
$arr = $newarr;
} while(strlen($arr[0])<4);
//in order to remain in the minority, we're only looking for scenarios in which no more than one other person also chooses blue
$c = 0;
foreach($arr as $a){
if(substr_count($a,'0')<=1){
$c++;
}
}
echo $c / count($arr); //0.3125
?>