## Let's Make a Deal AnswerYou're on the famous game show, and Monty Hall, the host, presents you with 3 closed doors. Behind one of the doors is a new Rolls Royce, but behind the other two are gag prizes. You choose one door, but before it is opened, Monty opens one of the unchosen doors, revealing a pile of manure. He invites you to choose again: either keep the door you have already chosen, or switch to the other, unknown door. What do you do, and why? Answer: You will DOUBLE your chances of winning if you SWITCH!The before-the-game probability that the prize will be behind any given door is 1/3. Assume that you choose door A, and that Monty opens door B. The probability that Monty opens B if the prize were behind A P(Monty opens B|A)=1/2 The probability that Monty opens B if the prize were behind B P(Monty opens B|B)=0 The probability that Monty opens B if the prize were behind C P(Monty opens B|C)=1 Then the probability Monty opens door B is then P(Monty opens B)=P(A)* P(M.o B|A)+ P(B)* P(M.o B|B) + P(C)* P(M.o B|C) =1/6 + 0 + 1/3 or 1/2 Then by Bayes Theorem: P(A|Monty opens B)=P(A)* P(Monty opens B|A)/P(Monty opens B) =(1/6) * (1/2) or 1/3P(C|Monty opens B)=P(C)* P(Monty opens B|C)/P(Monty opens B) =(1/3) * (1/2) or 2/3 |