The Monty Hall Problem

This problem is based on the game show, "Let's Make A Deal", which starred Monty Hall. There was much discussion of this problem in 1990 and 1991 in the column "Ask Marilyn" in Parade Magazine. Here is the problem: 

Suppose that you are 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 other doors, open another door, say No. 3, which has a goat. He then says to you, "Do you want to switch and pick door #2?" Is it to your advantage to switch?

 We will study this problem using probability.

 Total number of wins: _______ Total number of losses: ____________

Number of switch and win ____ Number of switch and lose __________

Number of stay and win ______ Number of stay and lose ____________

P(winning) = ________________ P(losing)=_________________________

P(winning | switch) = _______ P(winning | stay)= ________________

P(losing | switch) = ________ P(losing | stay)=__________________