The Monty Hall Problem Andr´e Reslow December 8, 2014 The Monty Hall problem is a probability puzzle based on the American television game show Let’s Make a Deal and named after its original host, Monty Hall.1 The Problem: 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? Intuition tells us (most people) that it should not matter if the contestant switches or not. It, is 50/50 to get the car. Or is it? I did a Matlab simulation of the problem2 . Some assumptions/rules: • The host must always open a door that was not picked by the contestant • The host must always open a door to reveal a goat and never the car • The host must always offer the chance to switch between the originally chosen door and the remaining closed door • The car is randomly places and the contestant have no prior information about placement of the door and assigns the same prior probability to each door The two strategies: • Stick with the door first chosen (No switch/Staying) • Change to the other door (Switching) I ran 100 000 simulations (”played games”) and got these results: Probability of winning if... • ...sticking with the first chosen door (No switch/Staying): 0.33 • ...changing door (Switching): 0.66 So, the best strategy is switching door! The ”explanation” is that three scenarios could occur, and the contestant will only lose if she picked the car in the first place, given the switching strategy. The three scenarios: • Contestant picks car. Host opens goat A or B. Switching is a losing strategy • Contestant picks goat A. Host opens goat B. Switching is winning strategy • Contestant picks goat B. Host opens goat A. Switching is winning strategy 1 http://en.wikipedia.org/wiki/Monty 2 Matlab

Hall problem code on my website: https://sites.google.com/site/andrereslow/research

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