91Ó°ÊÓ

Jerry Stackhouse of the National Basketball Association's Dallas Mavericks is the best freethrow shooter on the team, making \(89 \%\) of his shots (ESPN website, July, 2008 ). Assume that late in a basketball game, Jerry Stackhouse 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 Dallas Mavericks' 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 Dallas Mavericks' center is a better strategy than intentionally fouling Jerry Stackhouse.

Step by step solution

01

Determine the Probability of Both Shots Made by Stackhouse

Jerry Stackhouse makes 89% of his free-throw shots. The probability that he makes both shots is given by the product of the individual probabilities: \[ P(\text{both made}) = 0.89 \times 0.89 = 0.7921. \]

02

Determine the Probability of Stackhouse Making At Least One Shot

To solve this, first calculate the probability of him missing both shots, and then subtract it from 1.\[P(\text{both missed}) = (1 - 0.89) \times (1 - 0.89) = 0.11 \times 0.11 = 0.0121.\]Then the probability of making at least one shot is:\[P(\text{at least one made}) = 1 - P(\text{both missed}) = 1 - 0.0121 = 0.9879.\]

03

Determine the Probability of Stackhouse Missing Both Shots

As calculated in Step 2, the probability of Jerry Stackhouse missing both shots is:\[ P(\text{both missed}) = 0.0121. \]

04

Repeat Steps for the Center Making Both Shots

The center makes 58% of his shots. Therefore, the probability of making both is:\[ P(\text{both made}) = 0.58 \times 0.58 = 0.3364. \]

05

Calculate the Probability of the Center Making At Least One Shot

Calculate the probability of the center missing both shots:\[P(\text{both missed}) = (1 - 0.58) \times (1 - 0.58) = 0.42 \times 0.42 = 0.1764.\]Then the probability of making at least one shot is:\[P(\text{at least one made}) = 1 - P(\text{both missed}) = 1 - 0.1764 = 0.8236.\]

06

Calculate the Probability of the Center Missing Both Shots

As calculated in Step 5, the probability of the center missing both shots is:\[ P(\text{both missed}) = 0.1764. \]

07

Compare the Strategies

The probability of Stackhouse making both shots is 0.7921, while the center's is 0.3364. The probability of Stackhouse making at least one shot is 0.9879, while the center's is 0.8236. Thus, intentionally fouling the center is better since the likelihood of him missing both is higher at 0.1764 compared to Stackhouse's 0.0121.

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Key Concepts

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

Free Throw Shooting

Free throw shooting is a critical skill in basketball, often turning into a defining element during close games. It refers to the ability of a player to score points from unopposed attempts at the basket, typically allocated after a foul by the opposing team. Understanding free throw success involves grasping the concept of probability. For instance, if a player like Jerry Stackhouse has a free throw shooting percentage of 89%, it means he scores 89 times out of every 100 attempts. In the realm of probability, calculating the likelihood of different shooting outcomes, such as making both shots or missing both, is an important tool. This statistics-driven approach can provide insights into a player's consistency and reliability under pressure. Mastering this concept helps not only in pinpointing individual performance but also strategic decision-making for the team.

Basketball Statistics

Basketball statistics provide a quantitative means to gauge player performance and team strategies, offering a window into the effectiveness, strengths, and weaknesses of various players and tactics. Statistics such as shooting percentages, assists, rebounds, and turnovers serve as critical indicators. For players like Jerry Stackhouse, proficiency in free throws can be numerically expressed, for example, with an 89% success rate. This data allows coaches, teams, and analysts to evaluate his reliability in clutch situations.
Analyzing these stats, a coach might determine when to best leverage a player's skills or compensate for his weaknesses. Every stat is an opportunity to explore various aspects of the game, forming a foundational element in preparing winning strategies. Essentially, statistical data serves as a factual basis for everything from game-day decisions to training adjustments.

Sports Strategy

In basketball, sports strategy often revolves around exploiting your team’s strengths and the opponent's weaknesses. A common tactic is intentional fouling, particularly late in the game, to gain strategic advantage by stopping the clock and forcing the opposing player to rely on their free-throw skills. Targeting the weakest free throw shooters increases the chance of regaining possession while minimizing points gained by the opponent.
For example, as outlined in the original solution, a coach might choose to foul a player with only a 58% free throw success rate instead of Jerry Stackhouse, who hits 89%. By understanding these probabilities, teams can implement tactics that maximize their winning potential. Basketball strategy involves a careful balance of risk and opportunity, with statistical insights informing many critical in-game decisions.

Statistical Comparison

Statistical comparison allows teams to evaluate players and strategies through a numerical lens. By comparing the probabilities of making or missing free throws, teams can decide the best course of action during a game. In the example provided, Jerry Stackhouse's probabilities of making both free throws, making at least one shot, and missing both were calculated, as were those of the team's center.

• For Stackhouse, the probability of making both shots was 0.7921.
• Contrast this with the center's probability of 0.3364.
• Moreover, Stackhouse’s chance of making at least one shot stood at 0.9879, while the center's probability was 0.8236.
• Missing both shots was most likely for the center at 0.1764, compared to just 0.0121 for Stackhouse.

This statistical analysis confirms that fouling the center, with a lower shooting percentage, offers a greater strategic advantage. These comparisons are a cornerstone in optimizing basketball strategies, allowing for informed, evidence-based decisions in high-stakes environments.