Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two” rule: the first player to win two games more than his opponent wins the match. Find the expected number of games played.Hint: Consider the first two games as a pair, then the next two as a pair, etc.

Question

Yahir Wolfe · Accepted Answer

Step 1Given by is the expectation of a geometric random variable with parameter p by&amp;nbsp;1&amp;nbsp;p&amp;nbsp;. So in this case, we would have an expectation of&amp;nbsp;1&amp;nbsp;p&amp;nbsp;2&amp;nbsp;+&amp;nbsp;q&amp;nbsp;2&amp;nbsp;trials.Step 2But each trial corresponds to 2 games, so we get an expected value of&amp;nbsp;2&amp;nbsp;p&amp;nbsp;2&amp;nbsp;+&amp;nbsp;q&amp;nbsp;2&amp;nbsp;games played.

Rachael Trevino · Accepted Answer

Step 1Let N be the expected duration.The probability that Calvin wins is p, after which the expected duration is                                                                                                1                                                                                  ⏞                                                                step taken                                                          to get here                                            +                                                                                                p                                                    1                                                                                                                      ⏞                                                                probability                                                          of one more                                            +                                         (          1          −          p          )                           ⏞                                                                probability of                                                                                            N                  +                  1                                 more                                                          (  N  +  1  )Step 2Thus,                    N                            =        p        (        1        +        p        +        (        1        −        p        )        (        N        +        1        )        )        +        (        1        −        p        )        (        1        +        1        −        p        +        p        (        N        +        1        )        )                                          =        2        +        2        p        (        1        −        p        )        N                                          =                  2                      1            −            2            p            +            2                          p              2

Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two” rule: the first player to win two games more than his opponent wins the match. Find the expected number of games played.

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