Suppose it’s the 1970’s and you’re on the show Let’s Make a Deal. The host, Monty...

Suppose it’s the 1970’s and you’re on the show Let’s Make a Deal. The host, Monty Hall, would offer the “Big Deal” to a contestant at the end. In exchange for their current winnings, they would be able to choose one of three doors. Behind one of the doors is a very big prize that you would want (We will call this the “winning door”), and behind the other two doors is something you would not want. Whichever door you choose, there will always be at least one non-winning door that Monty Hall will then open and reveal that “It’s a good thing you didn’t choose that one!” (In the case you chose the winning door, the host would decide randomly which of the other two doors to open.) Then, Monty Hall will give you a decision to make: You may switch your choice to the other non-revealed door if you would like.

Now let "door A" be the name of the door you chose, and let the event A be that this is the winning door. Let B and C be the events that "door B" and "door C" are the winning doors, respectively. Which door the prize is placed behind is random, so your prior probabilities are P(A) = P(B) = P(C) =1/3. Also say "door B" is the name of whichever door that Monty Hall then decides to reveal. And let openB be the event that Monty hall revealed "door B".

(a) In a sentence or two for each, explain why P(openB|A) = 1/2, P(openB|B) = 0, and P(openB|C)= 1.

(b) Given the priors and the values from part a, use Bayes's Rule and the Law of Total Probability to calculate both P(A|openB) and P(C|openB)

(c) In a letter to The American Statistician in 1975 a version of the following problem was proposed and solved, and it a gained a considerable amount of fame after reappearing in a Parade Magazine column in 1990:

Do you want to keep door A, or do you want to switch to door C? Or does it not matter? Now that you have calculated the probabilities of the prize being behind these doors, explain which decision you would make.

Solutions

Expert Solution

Answer:

Given that:

suppose it's the 1970's and you 're on the show let's make a deal.The host monty hall,would offer the "Big Deal" to a contestant at the end.

(a)

P(open B/A) means A is the winning door and it is also chosen by the contestant, then Monty Hall can open any of the doors B or C. Therefore, Probability of opening door B out of the two available doors B and C shall be ½.
P(open B/B) means B is the winning door and the contestant has chosen door A. Now Monty Hall can open only door C, which does not have prize behind it, from the available doors B &C. Therefore, Probability of opening door B out of the two available doors B and C shall be 0.
P(open B/C) means C is the winning door and the contestant has chosen door A. Now Monty Hall can open only door B, which does not have prize behind it, from the available doors B &C. Therefore, Probability of opening door B out of the two available doors B and C shall be 1 as he has to open door B only as per the rules of the game.

b) From Baye 's rule,

Switching to door C is a better strategy because the probability of winning the prize with this strategy is 2/3 compared to the probability of winning the prize with the other strategy of keeping the door A which is 1/3. Explanation given below:
Strategy-1: Keep door A after door B, which does not contain the prize, has been opened by Monty Python.
Probability for Strategy-1: 1/3
The only way that the contestant can win by staying with the initial choice of door A is if the initial choice happened to be the door that has a prize behind it. And because the contestant is sticking with the initial choice, he can actually kind of forget about the rest of the game, about opening of the other door and about switching.
It's as if he is playing a simpler game, which is just that he have three doors, one of them has a prize behind it, and he choose one of them. Hence probability of this strategy winning the game is simply 1/3.
Strategy-2: Switch to door C after door B, which does not contain the prize, has been opened by Monty Python.
Probability for Strategy-2: 2/3
If the first choice happens to be the right door .i.e door A is the winning prize, then switching away from that door will always lose. But having the first choice as the winning door has a probability of one third. But for the rest of the time with probability 2/3, the first choice would be wrong.
Hence by switching the door, the probability of win will be 2/3.

orchestra answered 1 year ago

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