Goat or Car? Game Show Simulator
A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?
My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.
Player won 2 times and lost 8 times.
- Car
- Goat
- Goat
- Car
- Goat
- Goat
- Goat
- Goat
- Goat
- Goat
Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:
Player won 338 times and lost 662 times.
- Goat
- Car
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But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:
Player won 87 times and lost 913 times.
So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.
Would you like to run a simulation? Enter your variables below:
Custom test result
Player won 24 times and lost 76 times.
Methodology
This page uses the following PHP function to run the simulations:
function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){
// sanitize iterations
$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
$iterations = ($iterations < 1)? 10 : $iterations;
$iterations = ($iterations > 10000)? 10000 : $iterations;
// sanitize number_of_doors
$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;
// generate a standard array
$doors_original[0] = 'car';
for($i=1; $i < $number_of_doors ; $i++){
$doors_original[$i] = 'goat';
}
// initialize our scores
$losses = 0; $wins = 0;
// decide how big the images will be
if($iterations >= 1000){
$size = '24';
} else if($iterations >= 100){
$size = '32';
} else {
$size = '64';
}
$output = '<ol>';
for($i=0; $i<$iterations; $i++){
// make a temporary copy for this usage
$doors = $doors_original;
// shuffle the array
shuffle($doors);
// player always picks door number 1
$players_first_choice = $doors[0];
// did the player win?
if($players_first_choice == 'car' && $player_stays==true){
$output .= '<li class="c'.$size.'"><b>Car</b></li>';
$wins++;
} else if($players_first_choice == 'goat' && $player_stays==true){
$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
$losses++;
} else if($players_first_choice == 'car' && $player_stays==false){
$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
$losses++;
} else if($players_first_choice == 'goat' && $player_stays==false){
$output .= '<li class="c'.$size.'"><b>Car</b></li>';
$wins++;
}
}
$output .= '</ol><div class="clear"> </div>';
echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
if($show_pictures){
echo $output;
}
}