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Chapter 3: Problem 62

\(A, B\), and \(C\) are evenly matched tennis players. Initially \(A\) and \(B\) play a set, and the winner then plays \(C\). This continues, with the winner always playing the waiting player, until one of the players has won two sets in a row. That player is then declared the overall winner. Find the probability that \(A\) is the overall winner.

Short Answer

Expert verified
The probability that A is the overall winner is \(0.375\) or \(37.5\%\).

Step by step solution

01

Define The Probabilities

Each set has a 50% chance for each player to win, so let's denote this for our calculations. - Probability of A winning = \(P(A) = 0.5\) - Probability of B winning = \(P(B) = 0.5\) - Probability of C winning = \(P(C) = 0.5\)
02

Create a Probability Tree

Let's create a probability tree with the possible outcomes of the game. Start with the first set where A plays against B. 1. A vs. B: There are 2 possible outcomes-A wins or B wins. Now, we'll consider possible outcomes when the winner plays against C. 2. If A wins the first set: a. A vs. C: A wins or C wins b. If A wins two sets in a row, A is the overall winner. c. If C wins, the game proceeds with C vs. B. 3. If B wins the first set: a. B vs. C: B wins or C wins b. If B wins two sets in a row, B is the overall winner. c. If C wins, the game proceeds with C vs. A. Now that we have all the possible outcomes, we can derive a condition for A winning the game.
03

Define the Condition for A Winning

A can only win the game in one of the following scenarios: 1. A wins two sets in a row: A beats B and then A beats C. 2. C wins a set after B: C beats B and then A beats C.
04

Calculate the Probability for Each Scenario

1. Probability of A winning two sets in a row: - \(P(A \: wins \: two \: in \: a \: row) = P(A \: beats \: B) * P(A \: beats \: C) = 0.5 * 0.5 = 0.25\) 2. Probability of C winning a set after B and then A winning: - \(P(C \: wins \: after \: B \: and \: A \: wins) = P(A \: loses \: to \: B) * P(C \: beats \: B) * P(A \: beats\: C) = 0.5 * 0.5 * 0.5 = 0.125\)
05

Calculate the Total Probability for A Winning

Now let's sum up the probabilities from both scenarios: \(P(A \: is \: overall \: winner) = P(A \: wins \: two \: in \: a \: row) + P(C \: wins \: after \: B \: and \: A \: wins) = 0.25 + 0.125 = 0.375\) So, the probability that A is the overall winner is 0.375 or 37.5%.

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