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12. Bob and Alice enter an archery contest. Bob hits his target with probability P, p<1, and Alice hits her target, independently of Bob, with probability q, 0

12. Bob and Alice enter an archery contest. Bob hits his target with probability P, p<1, and Alice hits her target, independently of Bob, with probability q, 0

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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12. Bob and Alice enter an archery contest. Bob hits his target with probability p, 0< p<1, and Alice hits her target, independently of Bob, with probability q, 0 <q < 1. Bob and Alice both shoot one arrow at their respective targets. If they both hit their targets they stop. Otherwise, they shoot again, and they continue shooting, one round after another, until both hit their targets. (a) Let X count the number of times they have to shoot until they both hit their target. Identify the random variable X using the '' notation, and then write down its pmf with the correct parameters. X ~ Px (I) (b) What is the expected number of rounds until they both hit their targets? Give a formula involving p and q. x= (c) What is the probability that it takes at least three rounds for both of them to hit their respective targets? Give a formula involving p and q. 13
Transcribed Image Text:12. Bob and Alice enter an archery contest. Bob hits his target with probability p, 0< p<1, and Alice hits her target, independently of Bob, with probability q, 0 <q < 1. Bob and Alice both shoot one arrow at their respective targets. If they both hit their targets they stop. Otherwise, they shoot again, and they continue shooting, one round after another, until both hit their targets. (a) Let X count the number of times they have to shoot until they both hit their target. Identify the random variable X using the '' notation, and then write down its pmf with the correct parameters. X ~ Px (I) (b) What is the expected number of rounds until they both hit their targets? Give a formula involving p and q. x= (c) What is the probability that it takes at least three rounds for both of them to hit their respective targets? Give a formula involving p and q. 13
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