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Winning by a growing margin

EasyProbabilityBrainteasers~5m

Players A and B play rounds; A wins each round with probability 0.60.6 independently. They play until one player has won NN more rounds than the other, and that player wins the match. As N→∞N \to \infty, what does the probability that A wins the match approach?

Show hints (2)+
  1. The lead is a biased random walk; barriers at ±N\pm N give P[A]=11+(q/p)N\mathbb{P}[A]=\tfrac1{1+(q/p)^N}.
  2. q/p=23<1q/p=\tfrac23<1, so as N→∞N\to\infty this →1\to 1.

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Asked at: Optiver, SIG

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