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Let's Make a Deal Answer


You'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 1/3

P(C|Monty opens B)=P(C)* P(Monty opens B|C)/P(Monty opens B)
=(1/3) * (1/2) or 2/3