Win_odd Dhamnekar
Junior Member
- Joined
- Aug 14, 2018
- Messages
- 207
Example:
Mr. Goldsmith is 80 percent certain that his missing key is in one of the two pockets of his hanging jacket, being 40 percent certain it is in the left-hand pocket and 40 percent certain it is in the right-hand pocket.
If a search of the left-hand pocket does not find the key, what is the conditional probability that it is in the other pocket?
Solution: If we let L be the event that the key is in the left-hand pocket of the jacket, and R be the event that it is in the right-hand pocket, then the desired probability [imath] P[R|L^c][/imath] can be obtained as follows: [math] P[R|L^c]= \frac{P[RL^c]}{P[L^c]} = \frac{P[R]}{1-P[L]} = \frac23[/math]
Now, let us move to main question:
Considering the above example, 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 [imath]S_R[/imath] be the event that a search of the right- hand jacket pocket will be successful in finding the key, and let [imath] U_L[/imath] be the event that a search of the left-hand jacket pocket will be unsuccessful and, thus, not find the key. Find [imath] P[S_R|U_L] [/imath] 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 [math] P[S_R|U_L]= \frac{P[S_RU_L]}{P[U_L]}[/math]
Determining [imath]P[S_RU_L][/imath] by conditioning on whether or not the key is in the right-hand pocket, and determining [imath]P[U_L][/imath] by conditioning on whether or not the key is in the left- hand pocket;
(b) using the identity [math] P[S_R|U_L]= P[S_R|RU_L] P[R|U_L] + P[S_R|R^cU_L] P[R^c|U_L][/math]
Any math help will be accepted. I am working on this question. I have not made any progress as such to express here in writing mathematically.
Mr. Goldsmith is 80 percent certain that his missing key is in one of the two pockets of his hanging jacket, being 40 percent certain it is in the left-hand pocket and 40 percent certain it is in the right-hand pocket.
If a search of the left-hand pocket does not find the key, what is the conditional probability that it is in the other pocket?
Solution: If we let L be the event that the key is in the left-hand pocket of the jacket, and R be the event that it is in the right-hand pocket, then the desired probability [imath] P[R|L^c][/imath] can be obtained as follows: [math] P[R|L^c]= \frac{P[RL^c]}{P[L^c]} = \frac{P[R]}{1-P[L]} = \frac23[/math]
Now, let us move to main question:
Considering the above example, 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 [imath]S_R[/imath] be the event that a search of the right- hand jacket pocket will be successful in finding the key, and let [imath] U_L[/imath] be the event that a search of the left-hand jacket pocket will be unsuccessful and, thus, not find the key. Find [imath] P[S_R|U_L] [/imath] 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 [math] P[S_R|U_L]= \frac{P[S_RU_L]}{P[U_L]}[/math]
Determining [imath]P[S_RU_L][/imath] by conditioning on whether or not the key is in the right-hand pocket, and determining [imath]P[U_L][/imath] by conditioning on whether or not the key is in the left- hand pocket;
(b) using the identity [math] P[S_R|U_L]= P[S_R|RU_L] P[R|U_L] + P[S_R|R^cU_L] P[R^c|U_L][/math]
Any math help will be accepted. I am working on this question. I have not made any progress as such to express here in writing mathematically.
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