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Suppose that a basketball player's success in free-throw shooting can be described with a Markov chain. If the player made the last free throw, then she is four times more likely to make the next free throw as miss it. If the player missed her last free throw, then she is equally likely to make or miss the next free throw. 1. Find the transition matrix for the Markov chain. 2. If she made her last free throw, what is the probability she makes the next two in a row, for a total of three in a row? 3. If the she makes he third free throw, what is the probability she makes her fifth free throw? 4. If she misses her first free throw, what is the probability she also misses her third and fifth free throw?

Suppose that a basketball player's success in free-throw shooting can be described with a Markov chain. If the player made the last free throw, then she is four times more likely to make the next free throw as miss it. If the player missed her last free throw, then she is equally likely to make or miss the next free throw. 1. Find the transition matrix for the Markov chain. 2. If she made her last free throw, what is the probability she makes the next two in a row, for a total of three in a row? 3. If the she makes he third free throw, what is the probability she makes her fifth free throw? 4. If she misses her first free throw, what is the probability she also misses her third and fifth free throw?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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**Understanding the Markov Chain in Basketball Free-Throw Shooting** **Scenario:** Suppose that a basketball player’s success in free-throw shooting can be described with a Markov chain. The dynamics are as follows: - If the player made the last free throw, she is four times more likely to make the next free throw as miss it. - If the player missed her last free throw, she is equally likely to make or miss the next free throw. **Tasks:** 1. **Find the Transition Matrix for the Markov Chain.** - A transition matrix is used to describe the probabilities of transitioning from one state to another. Here, the states are whether the player makes or misses a free throw after making or missing the last one. 2. **Probabilities of Consecutive Makes:** - If she made her last free throw, determine the probability she makes the next two in a row, achieving a total of three successful free throws consecutively. 3. **Probability of Making the Fifth Free Throw:** - Given she makes her third free throw, what is the probability she successfully makes her fifth free throw? 4. **Probability of Consecutive Misses:** - If she misses her first free throw, calculate the probability she also misses her third and fifth free throws. **Visuals:** - The image contains boxes likely meant to hold numerical values or formulas representing these probabilities in terms of a matrix, although specific values are not filled in the text provided. This is a typical structure for organizing transitions in Markov chains into a matrix form which helps to compute the probabilities of interest.
Transcribed Image Text:**Understanding the Markov Chain in Basketball Free-Throw Shooting** **Scenario:** Suppose that a basketball player’s success in free-throw shooting can be described with a Markov chain. The dynamics are as follows: - If the player made the last free throw, she is four times more likely to make the next free throw as miss it. - If the player missed her last free throw, she is equally likely to make or miss the next free throw. **Tasks:** 1. **Find the Transition Matrix for the Markov Chain.** - A transition matrix is used to describe the probabilities of transitioning from one state to another. Here, the states are whether the player makes or misses a free throw after making or missing the last one. 2. **Probabilities of Consecutive Makes:** - If she made her last free throw, determine the probability she makes the next two in a row, achieving a total of three successful free throws consecutively. 3. **Probability of Making the Fifth Free Throw:** - Given she makes her third free throw, what is the probability she successfully makes her fifth free throw? 4. **Probability of Consecutive Misses:** - If she misses her first free throw, calculate the probability she also misses her third and fifth free throws. **Visuals:** - The image contains boxes likely meant to hold numerical values or formulas representing these probabilities in terms of a matrix, although specific values are not filled in the text provided. This is a typical structure for organizing transitions in Markov chains into a matrix form which helps to compute the probabilities of interest.
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