Problems in Probability

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Fall 2011 Homework 3: Question 6
Players A and B take turns in answering trivia questions, starting with player A answering the first question. Each time A answers a question, she has probability of getting it right. Each time B plays, he has probability of getting it right.
(a) If A answers m questions, what is the PMF of the number of questions she gets right?
(b) If A answers m times and B answers n times, what is the PMF of the total number of questions they get right (you can leave your answer as a sum)? Describe exactly when/whether this is a Binomial distribution.
(c) Suppose that the first player to answer correctly wins the game (with no predetermined maximum number of questions that can be asked). Find the probability that A wins the game.
Solution: (a) The r.v. is Bin(m, p1). (b) Let T be the total number of questions they get right. To get a total of k questions right, it must be that A got 0 and B got k, or A got 1 and B got k − 1, etc. These are disjoint events so the PMF is P(T = t) = Summation from j = 0 to k, where k goes from 0 to m + n, on (mCj) p1^j (1-p1)^(m-j) (nC[k-j]) p2^(k-j) (1-p2)^(n-(k-j)). This is the Bin(m + n, p) distribution if p1 = p2 = p, as shown in class (using the story for the Binomial, or using Vandermonde's identity). For p1 not equal to p2, it's not a Binomial distribution, since the trials have different probabilities of success; having some trials with one probability of success and other trials with another probability of success isn't equivalent to having trials with some "effective" probability of success. (c) Let r = P(A wins). Conditioning on the results of the first question for each player, we have r = p1/((1-p1)(1-p2)).
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
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