Calculating the Probability Outcomes of the Monty Hall Problem Using A Web-Based Simulation

Abstract

The Monty Hall Problem (MHP) has ignited a debate ever since it aired on television. The game lets a contestant choose one of three doors. Two of the doors contains a goat while the other one is a car. The host of the show will open one door that always contains a goat. After that, the host will ask the contestant if they want to switch their chosen door to the unopened last door. Most contestants will just stick to their chosen door but mathematical calculations show that switching doors will give contestants a 0.67 probability of winning. Mathematical and statistical methods to prove the probability of winning the game might not be enough for some people but the visual simulation of it might. This study created a simulation of the MHP that runs in a web browser where doors are chosen randomly. Through running different numbers of simulation from 10 to 100,000 game plays, the calculated probability that the car is at the chosen door is close to 0.33. The probability of the car is behind the last door is also close to 0.67. Same with the results of mathematical and statistical methods, the contestant should switch doors at the end. Aside from the three-door MHP, the simulation also tested 10-door and 100-door MHP. The results from all the simulations concluded that the probability of having the car is behind the chosen door is $\frac{1}{n}$ while the probability of having the car at the last door is $\frac{{n - 1}}{n}$ where is the number of doors.