Question

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Suppose that two teams (for fun, let’s call them the Domestic Shorthairs and Cache Cows) play a series of games to determine a winner. In a best-of-three series, the games end as soon as one team has won two games. In a best-of-five series, the games end as soon as one team has won three games, and so on. Assume that the Domestic Shorthair’s probability of winning any one game is p, where .5 < p < 1. (Notice that this means that the Domestic Shorthairs are the better team.) Also assume that the outcomes are independent from game to game.

We will compare different series configurations/rules on two criteria: the probability that the better team wins the series, and the expected value of the number of games needed to complete the series.

Best-of-Three Series

1. Determine (exactly) the probability that the Domestic Shorthairs win a best-of-three series, as a function of p. (Show your work. Also note that you solved this for a particular value of p in Quiz 4, so you might want to review that quiz.)

2. Graph this function for values of .5 < p < 1 (include good axis labels), and comment on its behavior. [Hint: As with Investigation 2, you could use Excel or RStudio or Wolfram Alpha to produce the graph.]

3.Let the random variable X3 = number of games played in a best-of-three series. Determine the probability distribution (pmf) of X3, as a function of p. (Show your work.)

Answer 1