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#2: Post edited
The author's solution doesn't expatiate why $\color{red}{P(C|X = 1) = P(C)}$?- >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:
- >
- >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
- >
- >![Image alt text](https://math.codidact.com/uploads/c2eDuHry2utfhsM2X9FpRFmk)
- Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
- p 17 in the publicly downloadable PDF of curbed solutions.
- The author's solution doesn't expatiate why $\color{red}{P(C|X = 1) = P(C)}$? [This similar question on Math Stack Exchange](https://math.stackexchange.com/q/4288685) has 0 answers, as at 4 January 2022.
- >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:
- >
- >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
- >
- >![Image alt text](https://math.codidact.com/uploads/c2eDuHry2utfhsM2X9FpRFmk)
- Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
- p 17 in the publicly downloadable PDF of curbed solutions.
#1: Initial revision
If C = Calvin wins the match, and $X \thicksim Bin(2, p) =$ how many of the first 2 games he wins — then why P(C|X = 1) = P(C)?
The author's solution doesn't expatiate why $\color{red}{P(C|X = 1) = P(C)}$? >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: > >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then > >![Image alt text](https://math.codidact.com/uploads/c2eDuHry2utfhsM2X9FpRFmk) Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94. p 17 in the publicly downloadable PDF of curbed solutions.