A Monte Carlo simulation of the Monty Hall problem

20 Aug 2025 - by 'Maurits van der Schee'

The Monty Hall problem is a probability puzzle. It has no logical contradiction, but for many people the result goes against their intuition. The problem originates from the American television game show "Let's Make a Deal". It is named after its original host, Monty Hall. The problem is stated as:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The problem can be simplified and summarized as:

If your initial guess is the car (chance 1/3) then you shouldn't switch, otherwise you should (and you will win).

To see that this is the case you can write down all possibilities or you can do what I did: a Monte Carlo simulation.

Monte Carlo in PHP

A Monte Carlo simulation is where you execute a probability experiment many times using a random number generator to determine the chances of winning. Here is the code written in PHP:

<?php
// function to determine win on monty hall problem
function monty_hall(array $doors, string $winner, string $chosen, bool $change): bool
{
    while (true) {
        // choose a random door to open
        $open = $doors[array_rand($doors)];
        // if door is not winning and not chosen open it
        if ($open != $winner && $open != $chosen) {
            break;
        }
    }
    // remove the open door from the array
    $doors = array_diff($doors, [$open]);
    // do you want to change?
    if ($change) {
        // remove the chosen door
        $doors = array_diff($doors, [$chosen]);
        // choose another door
        $chosen = $doors[array_rand($doors)];
    }
    return $chosen == $winner;
}
// monte carlo simulation to determine win chance of monty hall problem
$runs = 100000;
$wins = 0;
for ($i = 0; $i < $runs; $i++) {
    // doors is an array of closed doors
    $doors = ["Door 1", "Door 2", "Door 3"];
    // choose a random winning door
    $winner = $doors[array_rand($doors)];
    // choose a random door to open
    $chosen = $doors[array_rand($doors)];
    // increment win count on win
    if (monty_hall($doors, $winner, $chosen, true)) {
        $wins += 1;
    }
}
echo round(100 * $wins / $runs) . "%\n";

See: https://github.com/mevdschee/monty-hall-problem

Happy coding!

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TQ
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