Winning by a growing margin

EasyProbability Brainteasers~5m

Players A and B play rounds; A wins each round with probability $0.6$ independently. They play until one player has won $N$ more rounds than the other, and that player wins the match. As $N \to \infty$ , what does the probability that A wins the match approach?

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

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Winning by a growing margin

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