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Chapter 5: Problem 107

Two teams, \(\mathrm{A}\) and \(\mathrm{B}\), will play a best-of-seven series, which will end as soon as one of the teams wins four games. Thus, the series may end in four, five, six, or seven games. Assume that each team has an equal chance of winning each game and that all games are independent of one another. Find the following probabilities. a. Team A wins the series in four games. b. Team A wins the series in five games. c. Seven games are required for a team to win the series

Short Answer

Expert verified
The probabilities are: a. Team A wins the series in four games: \((0.5)^4\), b. Team A wins the series in five games: \(4 * (0.5)^5\), c. Seven games are required for a team to win the series: \(C(6, 3) * (0.5)^7\).

Step by step solution

01

Determine the Probability for Team A to Win in Four Games

In this scenario, Team A needs to win all four games outright, which can only happen in one way. Since the outcome of each game is independent and they have an equal chance of winning each game, the probability of winning a single game will be 0.5 (1/2). The events are independent hence the total probability will be the product of the probabilities of each game. Therefore, \(P(A wins in 4) = (0.5)^4\).
02

Determine the Probability for Team A to Win in Five Games

In this case, Team A must win four games and Team B must win one game. Furthermore, Team A's loss can occur in any of the first four games. The number of ways this can happen is 4 (Team A loses the first, second, third, or fourth game). Hence, the probability will be \(P(A wins in 5) = 4 * (0.5)^5\). Here, four represents the number of ways Team A can lose one of the first four games, and five represents the total number of games played.
03

Determine the Probability for a Team to Win in Seven Games

For a series to last seven games, each team must win three of the first six games. The number of ways of choosing 3 winning games for Team A out of 6 is given by the combination formula '6 choose 3'. So, we calculate \(C(6, 3)\) which equals 20. The probability of any specific sequence of three wins and three losses is \((0.5)^6\). As the team can win the final game, the total probability becomes: \(P(7 games played) = C(6, 3) * (0.5)^7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Best-of-Seven Series
A best-of-seven series is a competitive format where two teams face off until one team wins four games. This format is popular in various sports leagues, including NBA and NHL playoffs. The series only ends when a team wins four games, so it could last as few as four games and as many as seven games. In the case of two equally matched teams, the series is highly unpredictable due to the nature of individual game wins and losses. It offers excitement not only for fans but also challenges teams to adapt and strategize along the way.
Independent Events
In probability, independent events are those where the outcome of one event does not affect the outcome of another. This concept is crucial in calculating probabilities for scenarios like best-of-seven series, where each game's outcome doesn't influence the next. For example, if Team A wins the first game, it does not increase or decrease their chance of winning the next game. Understanding independence helps simplify probability calculations, as the joint probability of independent events is simply the product of their individual probabilities.
Binomial Probability
Binomial probability is a statistical method used when there are two possible outcomes for an event, such as winning or losing a game. In a best-of-seven series, calculating the probability of a certain series outcome uses the binomial probability formula. This formula requires understanding the number of trials (or games in this context), the success probability (probability of winning a game), and the desired number of wins. Each game has a probability of 0.5 due to equal chance, and the calculations for different series outcomes involve multiplying probabilities over multiple games, considering different scenarios of winning and losing across the games.
Combination Formula
The combination formula is used to determine how many ways certain outcomes can be achieved when order doesn't matter. It is often represented as \(C(n, r)\), where \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose. For example, to find how many ways a team can win three games out of six in a series, we use \(C(6, 3)\). This means we are selecting 3 games (wins) from a total of 6, and the combination formula helps identify all possible sequences of those events. Thus, combination formulas are key to calculating probabilities in series like the best-of-seven, where order of wins isn't as important as the total number.

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Most popular questions from this chapter

An office supply company conducted a survey before marketing a new paper shredder designed for home use. In the survey, \(80 \%\) of the people who used the shredder were satisfied with it. Because of this high acceptance rate, the company decided to market the new shredder. Assume that \(80 \%\) of all people who will use it will be satisfied. On a certain day, seven customers bought this shredder. a. Let \(x\) denote the number of customers in this sample of seven who will be satisfied with this shredder. Using the binomial probabilities table (Table I, Appendix C), obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Find the mean and standard deviation of \(x\). b. Using the probability distribution of part a, find the probability that exactly four of the seven customers will be satisfied.

On average, 20 households in 50 own answering machines. a. Using the Poisson formula, find the probability that in a random sample of 50 households, exactly 25 will own answering machines. b. Using the Poisson probabilities table, find the probability that the number of households in 50 who own answering machines is i. at most 12 ii. 13 to 17 iii. at least 30

Briefly explain the concept of the mean and standard deviation of a discrete random variable.

According to a 2011 poll, \(55 \%\) of Americans do not know that GOP stands for Grand Old Party (Time, October 17, 2011). Suppose that this result is true for the current population of Americans. a. Let \(x\) be a binomial random variable that denotes the number of people in a random sample of 17 Americans who do not know that GOP stands for Grand Old Party. What are the possible values that \(x\) can assume? b. Find the probability that exactly 8 people in a random sample of 17 Americans do not know that GOP stands for Grand Old Party. Use the binomial probability distribution formula.

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 20 per hour. The manager observes that 9 calls came into the mail-order company during a randomly selected 15 -minute period. a. If the rate of calls is actually 20 per hour, what is the probability that 9 or more calls will come in during a given 15 -minute period? b. If the rate of calls is really 30 per hour, what is the probability that 9 or more calls will come in during a given 15 -minute period? c. Based on the calculations in parts a and \(\mathrm{b}\), do you think that the rate of incoming calls is more likely to be 20 or 30 per hour? d. Would you advise the manager to hire a second operator? Explain.
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