5. Pooh’s Home Run Derby is a flash game supposedly made for children but notorious for its extreme difficulty. The objective of each level is simply to score a certain amount of home runs within a limited number of pitches. In the final level, you face Christopher Robin and must score at least 40 home runs in 50 pitches. Suppose that through hours of practice, you are able to score a home run on one of Christopher’s pitches with probability of 0.7. Assume each pitch is independent. Calculate:
(a) The probability of winning in exactly 40 rounds
(b) The probability of winning in exactly 41 rounds
(c) The probability of winning in exactly 42 rounds
(d) The probability of winning (hint: generalize the previous results and express your answer as a summation) Let X be a random variable for the total number of pitches it takes to win (for the purpose of simplicity, X may still map to a positive value greater than 50). Let W be a binary random variable returning 1 when we win and 0 when we lose (in a 50 pitch game). Please frame your answers in terms of these variables when applicable.
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