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Statistics / A First Course in Probability 9 / Chapter 3 / Problem 87P

A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross

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Consider Example 2a, but now suppose that when the key is

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QUESTION:

Problem 87P

Consider Example 2a, but now suppose that when the key is in a certain pocket, there is a 10 percent chance that a search of that pocket will not find the key. Let R and L be, respectively, the events that the key is in the right-hand pocket of the jacket and that it is in the left-hand pocket. Also, let SR be the event that a search of the right-hand jacket pocket will be successful in finding the key, and let UL be the event that a search of the left-hand jacket pocket will be unsuccessful and, thus, not find the key. Find P(SR|UL), the conditional probability that a search of the right-hand pocket will find the key given that a search of the left-hand pocket did not, by

(a) using the identity

determining P(SR|UL) by conditioning on whether or not the key is in the right-hand pocket, and determining P(UL) by conditioning on whether or not the key is in the left-hand pocket;

(b) using the identity

Questions & Answers

QUESTION:

Problem 87P

Consider Example 2a, but now suppose that when the key is in a certain pocket, there is a 10 percent chance that a search of that pocket will not find the key. Let R and L be, respectively, the events that the key is in the right-hand pocket of the jacket and that it is in the left-hand pocket. Also, let SR be the event that a search of the right-hand jacket pocket will be successful in finding the key, and let UL be the event that a search of the left-hand jacket pocket will be unsuccessful and, thus, not find the key. Find P(SR|UL), the conditional probability that a search of the right-hand pocket will find the key given that a search of the left-hand pocket did not, by

(a) using the identity

determining P(SR|UL) by conditioning on whether or not the key is in the right-hand pocket, and determining P(UL) by conditioning on whether or not the key is in the left-hand pocket;

(b) using the identity

ANSWER:

Step 1 of 2

There is a 10 percent chance that a search of that pocket will not find the key.

Let R and L be, respectively, the events that the key is in the right-hand pocket of the jacket and that it is in the left-hand pocket.

Also, let ${S}_{R}$ be the event that a search of the right-hand jacket pocket will be successful in finding the key, and let ${U}_{L}$ be the event that a search of the left-hand jacket will be unsuccessful and, thus not find the key.

We have to find the conditional probability P( ${S}_{R}/$ ${U}_{L}$ ) by using the identity,

P( ${S}_{R}/{U}_{L}$ ) = P( ${S}_{R}/R{U}_{L}$ ) P(R/ ${U}_{L}$ ) + P( ${S}_{R}/{{R}^{c}U}_{L}$ ) P( ${R}^{c}/{U}_{L}$ ).

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