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