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The probability you win a match against any opponent is p independently of the outcomes of all other matches. The probability that Gary K. wins a match against any opponent is q independently of the outcomes of all other matches. In the competition you will be playing consecutive matches against di fferent opponents until you lose for the fi rst time. Then all the oppponents you played against will play one game each against Gary K. Compute the expected number of matches Gary K. will win. Use the Markov inequality to bound the probability that he wins at least one match.

So far I can see this is geometric with parameter p. When I lose I stop and the number of matches I won would be k-1 wins , since I lost on my kth turn. Now Gary has k-1 opponents he could win against with probability q and lose again with probability 1-q . I am not sure what to do with this information. I understand the expected amount of matches gary wins will be related to how many I am expected to win, but I am not sure how to relate the two and through what equation.

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Let $G$ be the number of matches Gary wins, and let $Y$ be the number you win. We want $E(G)$. This is $E(E(G|Y))$. (I am not sure your course uses this notation. For details, see the Law of Total Expectation.)

Use the appropriate result about the expectation of a binomially distributed random variable to calculate $E(G|Y=y)$. Then use the geometric distribution probabilities to calculate $E(E(G|Y))$.

I hope you can comfortably finish things. If you indicate in a message that you have some difficulties, I can give some computational details.

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