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#2: Post edited by user avatar DNB‭ · 2022-01-17T08:54:19Z (about 2 years ago)
  • 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.
  • >![Image alt text](https://math.codidact.com/uploads/8fzNn3WLtxx7EMJioUZ95vPF)
  • >50. 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
  • >
  • >![Image alt text](https://math.codidact.com/uploads/F19HkT6h5i7s81At9y3dTt8m)
  • Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
  • p 18 in the publicly downloadable PDF of curbed solutions.
  • 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.
  • >![Image alt text](https://math.codidact.com/uploads/8fzNn3WLtxx7EMJioUZ95vPF)
  • >50. 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
  • >
  • >![Image alt text](https://math.codidact.com/uploads/F19HkT6h5i7s81At9y3dTt8m)
  • Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
  • p 18 in the publicly downloadable PDF of curbed solutions.
#1: Initial revision by user avatar DNB‭ · 2021-12-31T08:41:57Z (about 2 years ago)
Why's a 2-player game with "win by two" rule = Gambler's Ruin where each player starts with $2? Why N = 4?
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.

>![Image alt text](https://math.codidact.com/uploads/8fzNn3WLtxx7EMJioUZ95vPF)

>50. 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
>
>![Image alt text](https://math.codidact.com/uploads/F19HkT6h5i7s81At9y3dTt8m)

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