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Chapter 4: Problem 36

Reggie Miller of the Indiana Pacers is the National Basketball Association's best career free throw shooter, making \(89 \%\) of his shots \((U S A \text { Today, January } 22,2004\) ). Assume that late in a basketball game, Reggie Miller is fouled and is awarded two shots. a. What is the probability that he will make both shots? b. What is the probability that he will make at least one shot? c. What is the probability that he will miss both shots? d. Late in a basketball game, a team often intentionally fouls an opposing player in order to stop the game clock. The usual strategy is to intentionally foul the other team's worst free throw shooter. Assume that the Indiana Pacers' center makes \(58 \%\) of his free throw shots. Calculate the probabilities for the center as shown in parts (a), (b), and (c), and show that intentionally fouling the Indiana Pacers' center is a better strategy than intentionally fouling Reggie Miller.

Short Answer

Expert verified
Fouling the Pacers' center is a better strategy, as he is more likely to miss both shots (17.64% vs 1.21%) and less likely to make both (33.64% vs 79.21%).

Step by step solution

01

Understanding the Problem

In this problem, we are asked to find probabilities related to basketball free throws. Specifically, we will calculate probabilities for Reggie Miller, who has a success rate of making a free throw at 89%, and compare them to the Indiana Pacers' center, who has a success rate of 58%. The scenarios include making both shots, making at least one shot, and missing both shots.
02

Calculate Probability of Making Both Shots (Reggie Miller)

Reggie Miller's probability of making one free throw is 0.89. The probability of making both shots is calculated by multiplying the probabilities of making each shot independently: \[ P( ext{both made}) = 0.89 \times 0.89 = 0.7921 \] The probability that Reggie Miller will make both shots is 79.21%.
03

Calculate Probability of Making At Least One Shot (Reggie Miller)

We can find the probability of making at least one shot by subtracting the probability of missing both shots from 1. The probability of missing one shot is 1 - 0.89 = 0.11.Thus, the probability of missing both shots is: \[ P( ext{both missed}) = 0.11 \times 0.11 = 0.0121 \]Therefore, \[ P( ext{at least one made}) = 1 - 0.0121 = 0.9879 \] The probability that Reggie Miller will make at least one shot is 98.79%.
04

Calculate Probability of Missing Both Shots (Reggie Miller)

As calculated in Step 2, the probability of Reggie missing both shots is: \[ P( ext{both missed}) = 0.0121 \] Hence, Reggie Miller has a 1.21% chance of missing both shots.
05

Calculate Probabilities for the Pacers' Center (58% Chance of Making a Shot)

Let's calculate the same scenarios for the Pacers' center, who makes 58% of his shots. - **Probability of making both shots:** \[ P( ext{both made}) = 0.58 \times 0.58 = 0.3364 \] - **Probability of making at least one shot:** First, find the probability of missing both shots: \[ P( ext{both missed}) = (1 - 0.58)^2 = 0.42 \times 0.42 = 0.1764 \] \[ P( ext{at least one made}) = 1 - 0.1764 = 0.8236 \] - **Probability of missing both shots:** \[ P( ext{both missed}) = 0.1764 \]
06

Compare Strategies

Compare the calculated probabilities for Reggie Miller and the Pacers' center: For Reggie Miller: - Make both shots: 79.21% - Miss both shots: 1.21% - Make at least one shot: 98.79% For the Pacers' center: - Make both shots: 33.64% - Miss both shots: 17.64% - Make at least one shot: 82.36% The Pacers' center has a higher probability of missing both shots (17.64% vs 1.21%) and a lower probability of making both (33.64% vs 79.21%). Therefore, intentionally fouling the Pacers' center is a better strategy.

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Key Concepts

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

Binomial Probability
Binomial probability is a key concept in understanding how often an outcome occurs over a series of attempts, especially in sports. It's a great way to model situations like free throw shooting where there are two possible results: a made shot or a missed one. When discussing binomial probability, two important numbers come into play: the number of trials (shots taken) and the probability of one success (making a shot).

For a player like Reggie Miller, with an 89% success rate per shot, you can calculate the probability of making two consecutive shots. This process involves multiplying the probabilities of making each shot independently. Binomial probability can be used to calculate all kinds of outcomes in sports, from the likelihood of a basketball player making multiple shots in a row to a baseball player hitting a certain number of home runs in a season.
Statistical Analysis in Basketball
In basketball, much like other sports, statistical analysis helps in making informed decisions. Knowing a player's stats, like Reggie Miller's 89% free throw success rate, gives teams an edge in strategizing. These statistics aren't just random numbers; they tell a story about a player's performance over time.

By using statistical analysis, coaches and players can identify strengths and weaknesses. If a player like Miller has a high free throw percentage, it's a statistical insight that can affect game tactics. Beyond shooting, statistical analysis in basketball tracks various metrics like rebounds, assists, and blocks, offering a comprehensive view of a player's overall impact on the game.
Probability Calculations
Probability calculations in sports, particularly basketball, help quantify the chances of different outcomes. For instance, calculating the probability of Reggie Miller making at least one or both of his free throws involves understanding basic probability rules.

The formula for calculating probability is simple: multiply the likelihoods of each event. In Miller's case, multiplying his 89% chance of making one shot by itself gives the probability of making both. To find the chance of making at least one shot, you calculate the probability of missing both and subtract it from 1. This mathematical approach allows teams to predict game outcomes more accurately.
Decision Making in Sports
Decision making in sports is often guided by statistical and probability insights. Coaches use these insights to devise strategies that increase their chances of winning. In the scenario presented, understanding the statistical traits of players like Reggie Miller and the Pacers' center is crucial.

By choosing to foul a player with a lower free throw percentage, teams exploit an advantage. The Pacers' center, with a 58% success rate, is statistically more likely to miss than Reggie Miller. Decisions like these, while influenced by numbers, also consider the game situation and psychological factors. Smart decision making combines all these elements to optimize outcomes on the court.

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Most popular questions from this chapter

A consulting firm submitted a bid for a large research project. The firm's management initially felt they had a \(50-50\) chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for \(75 \%\) of the successful bids and \(40 \%\) of the unsuccessful bids the agency requested additional information. a. What is the prior probability of the bid being successful (that is, prior to the request for additional information)? b. What is the conditional probability of a request for additional information given that the bid will ultimately be successful? c. Compute the posterior probability that the bid will be successful given a request for additional information.

High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. The University of Pennsylvania received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admissions pool for further consideration. In the past, Penn has admitted about \(18 \%\) of the deferred early admission applicants during the regular admission process. Counting the additional students who were admitted during the regular admission process, the total class size was 2375 (USA Today, January 24,2001 ). Let \(E, R,\) and \(D\) represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. a. Use the data to estimate \(P(E), P(R),\) and \(P(D)\) b. Are events \(E\) and \(D\) mutually exclusive? Find \(P(E \cap D)\) c. For the 2375 students admitted to Penn, what is the probability that a randomly selected student was accepted for early admission? d. Suppose a student applies to Penn for early admission. What is the probability the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

A decision maker subjectively assigned the following probabilities to the four outcomes of an experiment: \(P\left(E_{1}\right)=.10, P\left(E_{2}\right)=.15, P\left(E_{3}\right)=.40,\) and \(P\left(E_{4}\right)=.20 .\) Are these probability assignments valid? Explain.

A 2001 preseason NCAA football poll asked respondents to answer the question, "Will the Big Ten or the Pac-10 have a team in this year's national championship game, the Rose Bowl?" Of the 13,429 respondents, 2961 said the Big Ten would, 4494 said the Pac- 10 would, and 6823 said neither the Big Ten nor the Pac- 10 would have a team in the Rose Bowl (http://www.yahoo.com, August 30, 2001). a. What is the probability that a respondent said neither the Big Ten nor the Pac-10 would have a team in the Rose Bowl? b. What is the probability that a respondent said either the Big Ten or the Pac-10 would have a team in the Rose Bowl? c. Find the probability that a respondent said both the Big Ten and the Pac-10 would have a team in the Rose Bowl.

The prior probabilities for events \(A_{1}\) and \(A_{2}\) are \(P\left(A_{1}\right)=.40\) and \(P\left(A_{2}\right)=.60 .\) It is also known that \(P\left(A_{1} \cap A_{2}\right)=0 .\) Suppose \(P\left(B \text { ? } A_{1}\right)=.20\) and \(P\left(B | A_{2}\right)=.05\) a. Are \(A_{1}\) and \(A_{2}\) mutually exclusive? Explain. b. Compute \(P\left(A_{1} \cap B\right)\) and \(P\left(A_{2} \cap B\right)\) c. Compute \(P(B)\) d. Apply Bayes' theorem to compute \(P\left(A_{1} | B\right)\) and \(P\left(A_{2} | B\right)\).
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