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Two basketball players take turns trying to throw the ball through the hoop until one of the them succeeds. In detail, the first player throws the ball first: if the ball falls through the hoop, the game ends, but if not, it's the turn of the second player. Now if the second player makes the ball fall through the hoop, the game ends. If not, then the first player throws again, etc. During each throw, the first player succeeds with probability p1, while the second one with probability p2. Assume these probabilities satisfy 0 < P1 + P2. Let X1 be the number of throws made by player 1. Find the probability mass function of X1. Verify that it is a probability mass function.

Two basketball players take turns trying to throw the ball through the hoop until one of the them succeeds. In detail, the first player throws the ball first: if the ball falls through the hoop, the game ends, but if not, it's the turn of the second player. Now if the second player makes the ball fall through the hoop, the game ends. If not, then the first player throws again, etc. During each throw, the first player succeeds with probability p1, while the second one with probability p2. Assume these probabilities satisfy 0 < P1 + P2. Let X1 be the number of throws made by player 1. Find the probability mass function of X1. Verify that it is a probability mass function.

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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plz provide E|x1| and E|x2|

Problem 1 (continuation of previous Problem 5): Two basketball players take turns trying to throw the ball through the hoop until one of the them succeeds. In detail, the first player throws the ball first: if the ball falls through the hoop, the game ends, but if not, it's the turn of the second player. Now if the second player makes the ball fall through the hoop, the game ends. If not, then the first player throws again, etc. During each throw, the first player succeeds with probability p1, while the second one with probability p2. Assume these probabilities satisfy 0 < p1 + P2. Let X; be the number of throws made by player i, where i E {1,2}. (i) Find E[X1]. Is it always finite? (ii) Find the probability mass function of X2. Verify that it is a probability mass function. Find E[X2].
Transcribed Image Text:Problem 1 (continuation of previous Problem 5): Two basketball players take turns trying to throw the ball through the hoop until one of the them succeeds. In detail, the first player throws the ball first: if the ball falls through the hoop, the game ends, but if not, it's the turn of the second player. Now if the second player makes the ball fall through the hoop, the game ends. If not, then the first player throws again, etc. During each throw, the first player succeeds with probability p1, while the second one with probability p2. Assume these probabilities satisfy 0 < p1 + P2. Let X; be the number of throws made by player i, where i E {1,2}. (i) Find E[X1]. Is it always finite? (ii) Find the probability mass function of X2. Verify that it is a probability mass function. Find E[X2].
Problem 5 Two basketball players take turns trying to throw the ball through the hoop until one of the them succeeds. In detail, the first player throws the ball first: if the ball falls through the hoop, the game ends, but if not, it's the turn of the second player. Now if the second player makes the ball fall through the hoop, the game ends. If not, then the first player throws again, etc. During each throw, the first player succeeds with probability p1, while the second one with probability p2. Assume these probabilities satisfy 0 < P1 + P2. Let X1 be the number of throws made by player 1. Find the probability mass function of X1. Verify that it is a probability mass function.
Transcribed Image Text:Problem 5 Two basketball players take turns trying to throw the ball through the hoop until one of the them succeeds. In detail, the first player throws the ball first: if the ball falls through the hoop, the game ends, but if not, it's the turn of the second player. Now if the second player makes the ball fall through the hoop, the game ends. If not, then the first player throws again, etc. During each throw, the first player succeeds with probability p1, while the second one with probability p2. Assume these probabilities satisfy 0 < P1 + P2. Let X1 be the number of throws made by player 1. Find the probability mass function of X1. Verify that it is a probability mass function.
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