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Unformatted text preview: Math 55, Spring 2006: Midterm #2, 6 April 2006 Problem 1: Consider the following variant of a wellknown game. Of three doors, one is secretly a winner (with equal probability). You choose a door to start. If your door is the winner, the host opens one of the others, showing it is empty, and offers you the choice to switch or stay. If your door is not the winner, the host either ends the game (with conditional probability 1 p ) or opens one of the others, showing it is empty, and offers the choice to switch or stay. (a) Describe the sample space. Do all the points in the sample space have the same probability? (b) Compute the probability that the host offers you the choice to switch or stay. (c) Compute the conditional probability that switching wins, given that you are offered the choice. (d) For what value of p does this game become the usual Monty Hall problem and what does your analysis then suggest? Solution: (a) Number the doors from (say) 0 to 2 and use two Boolean variables to indicate whether you are offered the choice to switch and whether you accept. Then the sample space is S = Z 3 × B × B × Z 3 , where the four entries of each point x = ( x 1 ,x 2 ,x 3 ,x 4 ) = ( p,c,a,w ) ∈ S are x 1 = p ∈ Z 3 for the door you pick, x 2 = c ∈ B for the offering of the choice, x 3 = a ∈ B for accepting the choice, and x 4 = w ∈ Z 3 for the number of the winning door. Since some possible combinations never occur, they have zero probability, so the points do not all have equal probabilities. Smaller sample spaces can be used if they describe all the details of the experiments, but they will complicate the calculations....
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 Fall '08
 STRAIN
 Math, Conditional Probability, Probability, Probability theory, Recurrence relation, monty hall problem, nonnegative integer solutions

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