Monty Hall Problem with Four Doors. Problem: Consider the following variation of
the car-goat problem solved in class. This time there are four doors, three ...
Monty Hall Problem with Four Doors
Problem: Consider the following variation of the car-goat problem solved in class. This time there are four doors, three goats, and one car. You choose a door at random and then the host selects a door with a goat behind it at random, which he opens. Suppose you switch to one of the other two doors, picking one at random. What is the probability now of winning the car? What is the probability of winning the car if you don’t switch? Solution: Without loss of generality, we assume that the car is behind door 1. (a) Suppose you decide to switch. Following the notational convention used in class, one can write the sample space of the game as
S = {(1, 2, 3, L), (1, 2, 4, L), (1, 3, 2, L), (1, 3, 4, L), (1, 4, 2, L), (1, 4, 3, L), (2, 3, 1,W ), (2, 3, 4, L), (2, 4, 1,W )(2, 4, 3, L), (3, 2, 1,W ), (3, 2, 4, L), (3, 4, 1,W ), (3, 4, 2, L), (4, 2, 1,W ), (4, 2, 3, L), (4, 3, 1,W ), (4, 3, 2, L)}. Since you pick the first door randomly, we get P[(1, 2, 3, L), (1, 2, 4, L), (1, 3, 2, L), (1, 3, 4, L), (1, 4, 2, L), (1, 4, 3, L)] =
1 4
and so you always lose if you pick door 1 first. If you pick door 2 first, we have 1 P[(2, 3, 1,W ), (2, 3, 4, L), (2, 4, 1,W ), (2, 4, 3, L)] = . 4 Further, since the host picks the door containing the goat randomly and you switch to a door randomly, we have P[(2, 3, 1,W )] = P[(2, 3, 4, L)] = P[(2, 4, 1,W )] = P[(2, 4, 3, L)] =
1 . 16
Similarly, we have P[(3, 2, 1,W )] = P[(3, 2, 4, L)] = P[(3, 4, 1,W )] = P[(3, 4, 2, L)] =
1
1 16
and P[(4, 2, 1,W )] = P[(4, 2, 3, L)] = P[(4, 3, 1,W )] = P[(4, 3, 2, L)] =
1 . 16
Therefore, P[You win] = P[(2, 3, 1,W )] + P[(2, 4, 1,W )] + P[(3, 2, 1,W )] + P[(3, 4, 1,W )] + P[(4, 2, 1,W )] + P[(4, 3, 1,W )] 1 3 = 6× = . 16 8 (b) Suppose you decide not to switch, the only way you can win the car is when your initial choice is door 1. So the answer is 1/4 or 2/8. Since 3/8 > 2/8, switching will therefore give you a slight advantage.
2
