(15 points) The Monte Hall Problem: You are on the game show Let\'s Make a Deal,
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(15 points) The Monte Hall Problem: You are on the game show Let's Make a Deal, hosted by Monte Hall. A new car was placed behind one of three doors. You select a door, and for simplicity you choose door 1. Before opening your door, Monte shows you what was behind one of the other two doors, let's say door number 3 (nothing) Finally, Monte asks yo like to switch your choice to door 2. What do you do? Use Bayes Theorem to justify your answer Hint u if you would like to keep your original door or if you would Let C be the door that contains the car, so P(C = 1) = P (C = 2) = P(C 3) =- Let S be the door Monte shows you. We can assume that under no circumstances would he show you the door with the car. Hence, we can model S given which door the car is actually behind as follows: 0 0 0 0.5 0 0.5 0 For example, the 1 in the bottom row of the table means that the probability Monte shows you what is behind door 2 (S) given that the car is actually behind door 3 (C) is 1. This is because Monte cannot show you what is behind door 1 since you picked it and he cannot show you door 3 either since the car is actually behind itExplanation / Answer
Assume we pick Door 1 and then Monty shows us a goat behind Door 2. Now let A be the event that the car is behind Door 1 and B be the event that Monty shows us a goat behind Door 2. Then
Pr(AB)=Pr(BA)×Pr(A)/Pr(B)=1/2×1/31/3×1/2+1/3×0+1/3×1=1/3..The tricky calculation is Pr(B)Pr(B). Remember, we are assuming we initially chose Door 1. It follows that if the car is behind Door 1, Monty will show us a goat behind Door 2 half the time. If the car is behind Door 2, Monty never shows us a goat behind Door 2. Finally, if the car is behind Door 3, Monty shows us a goat behind Door 2 every time. Thus,Pr(B)=1/3×1/2+1/3×0+1/3×1=1/2.The car is either behind Door 1 or Door 3, and since the probability that it's behind Door 1 is 1/3 and the sum of the two probabilities must equal 1, the probability the car is behind Door 3 is 11/3=2/3 You could also apply Bayes' Theorem directly, but this is simpler.
So Bayes says we should switch, as our probability of winning the car jumps from 1/3 to 2/3.
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