Chapter 3, Problem 3.91P

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 $S_R$ 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 pocket will be unsuccessful and, thus, not find the key. Find $P\left(S_R|U_L\right)$, 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 $P\left(S_R|U_L\right)=\frac{P\left(S_R{U}_L\right)}{P\left(U_L\right)}$ determining $P\left(S_R{U}_L\right)$ by conditioning on whether or not the key is in the right-hand pocket, and determining $P\left(U_L\right)$ by conditioning on whether or not the key is in the left-hand pocket;

b. using the identity $P\left(S_R|U_L\right)=P\left(S_R|R{U}_L\right)P\left(R|U_L\right)+P\left(S_R|R^C{U}_L\right)P\left(R^c{U}_L\right)$