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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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Most popular questions from this chapter

An urn contains three white, six red, and five black balls. Six of these balls are randomly selected from the urn. Let \(X\) and \(Y\) denote respectively the number of white and black balls selected. Compute the conditional probability mass function of \(X\) given that \(Y=3\). Also compute \(E[X \mid Y=1]\).

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\(A\) and \(B\) roll a pair of dice in turn, with \(A\) rolling first. A's objective is to obtain a sum of 6 , and \(B\) 's is to obtain a sum of 7 . The game ends when either player reaches his or her objective, and that player is declared the winner. (a) Find the probability that \(A\) is the winner. (b) Find the expected number of rolls of the dice. (c) Find the variance of the number of rolls of the dice.

The opponents of soccer team \(\mathrm{A}\) are of two types: either they are a class 1 or a class 2 team. The number of goals team A scores against a class \(i\) opponent is a Poisson random variable with mean \(\lambda_{i}\), where \(\lambda_{1}=2, \lambda_{2}=3\). This weekend the team has two games against teams they are not very familiar with. Assuming that the first team they play is a class 1 team with probability \(0.6\) and the second is, independently of the class of the first team, a class 1 team with probability \(0.3\), determine (a) the expected number of goals team A will score this weekend. (b) the probability that team \(\mathrm{A}\) will score a total of five goals.

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