A sequence of independent trials, each of which results in outcome number i with probability Pi , i = 1, . . . , n,? ? is observed until the same outcome occurs k times in a row; this outcome then is declared to be the winner of the game. For instance, if k = 2 and the sequence of outcomes is 1, 2, 4, 3, 5, 2, 1, 3, 3, then we stop after nine trials and declare outcome number 3 the winner. What is the probability that i wins, i = 1, . . . , n, and what is the expected number of trials?
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