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Why's a 2-player game with "win by two" rule = Gambler's Ruin where each player starts with $2? Why N = 4?

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  1. Why can this problem "be thought of as a gambler's ruin where each player starts out with $2"?

  2. Please see my red arrow. Why is this exponent 4? I quote op. cit. p 73.

Example 2.7.3 (Gambler's ruin). Two gamblers, A and B, make a sequence of \$1 bets. In each bet, gambler A has probability p of winning, and gambler B has probability $q = 1-p$ of winning. Gambler A starts with $i$ dollars and gambler B starts with $N-i$ dollars; the total wealth between the two remains constant since every time A loses a dollar, the dollar goes to B, and vice versa.

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  1. 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 probability that Calvin wins the match (in terms of p), in two different ways:

(a) by conditioning, using the law of total probability.

(b) by interpreting the problem as a gambler's ruin problem.

Solution:

(b) The problem can be thought of as a gambler's ruin where each player starts out with $2. So the probability that Calvin wins the match is

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Blitzstein, Introduction to Probability (2019 2 edn), Chapter 2, Exercise 50, p 94.
p 18 in the publicly downloadable PDF of curbed solutions.

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