The famous Monte Hall Problem poses the following question: “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 #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors?”

While the intuitive answers seems to be “no,” as one might argue that the two remaining doors are equally likely to contain the car, the correct answer is actually “yes.” As vos Savant points out later on in the above link, the probability of winning if you switch is actually 2/3.

But what if you wanted to find the solution without using probability directly? One way is through a Monte Carlo Simulation, which involves running a simulation of the game numerous times in order to calculate the probabilities of winning heuristically. The idea is that as the number of observations increases, the average of the results will coincide with the expected value.

For instance, if we run the simulation 1,000 times, we see a fair amount of volatility in the results over the first 250 and even 500 trials. As we add more trials, however, the average of the results begin to converge to the true expected values: 2/3 chance of winning if we switch doors, and 1/3 chance if we don’t.

The results are even more concrete if we consider 10,000 trials:

And there you have it, a simple application of Monte Carlo Simulation to support one of the more counter-intuitive results in probability theory.