# Suppose it’s the 1970’s and yo

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

Now let “door A” be the name of the door you chose, and let theevent A be that this is the winning door. Let B and C be the eventsthat “door B” and “door C” are the winning doors, respectively.Which door the prize is placed behind is random, so your priorprobabilities are P(A) = P(B) = P(C) =1/3. Also say “door B” is thename of whichever door that Monty Hall then decides to reveal. Andlet 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’sRule 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 versionof the following problem was proposed and solved, and it a gained aconsiderable amount of fame after reappearing in a Parade Magazinecolumn 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 theprobabilities of the prize being behind these doors, explain whichdecision you would make.

Given that:

suppose it’s the 1970’s and you ‘re on the show let’s make adeal.The host monty hall,would offer the “Big Deal” to a contestantat the end.

(a)

• P(open B/A) means A is the winning door and it is also chosenby the contestant, then Monty Hall can open any of the doors B orC. Therefore, Probability of opening door B out of the twoavailable doors B and C shall be ½.
• P(open B/B) means B is the winning door and the contestant haschosen door A. Now Monty Hall can open only door C, which does nothave prize behind it, from the available doors B &C. Therefore,Probability of opening door B out of the two available doors B andC shall be 0.
• P(open B/C) means C is the winning door and the contestant haschosen door A. Now Monty Hall can open only door B, which does nothave prize behind it, from the available doors B &C. Therefore,Probability of opening door B out of the two available doors B andC shall be 1 as he has to open door B only as per the rules of thegame.

b) From Baye ‘s rule,

C)

• Switching to door C is a better strategy because theprobability of winning the prize with this strategy is 2/3 comparedto the probability of winning the prize with the other strategy ofkeeping the door A which is 1/3. Explanation given below:
• Strategy-1: Keep door A after door B, which does not containthe prize, has been opened by Monty Python.
• Probability for Strategy-1: 1/3
• The only way that the contestant can win by staying with theinitial choice of door A is if the initial choice happened to bethe door that has a prize behind it. And because the contestant issticking with the initial choice, he can actually kind of forgetabout the rest of the game, about opening of the other door andabout switching.
• It’s as if he is playing a simpler game, which is just that hehave three doors, one of them has a prize behind it, and he chooseone of them. Hence probability of this strategy winning the game issimply 1/3.
• Strategy-2: Switch to door C after door B, which does notcontain 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 isthe winning prize, then switching away from that door will alwayslose. But having the first choice as the winning door has aprobability of one third. But for the rest of the time withprobability 2/3, the first choice would be wrong.
• Hence by switching the door, the probability of win will be2/3.

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