Oscar has lost his dog in either forest A (with a priori probability 0.4) or in

Question

Oscar has lost his dog in either forest A (with a priori probability 0.4) or in forest B (with a priori probability 0.6). On any given day, if the dog is in A and Oscar spends a day searching for it in A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog is in B and Oscar spends a day looking for it there, the conditional probability that he will find the dog that day is 0.15. The dog cannot go from one forest to the other. Oscar can search only in the daytime, and he can travel from one forest to the other only at night. (a) In which forest should Oscar look to maximize the probability he finds his dog on the first (b) Given that Oscar looked in A on the first day but didn't find his dog, what is the probability (c) If Oscar flips a fair coin to determine where to look on the first day and finds the dog on day of the search? that the dog is in A? the first day, what is the probability that he looked in A?

Explanation / Answer

Let the events be denoted as follows:

A

the dog is in forest A

B

the dog is in forest B

Dn

the dog dies on the nth evening

SAn

Oscar searches in forest A on the nth day

SBn

Oscar searches in forest B on the nth day

Fn

Oscar finds the dog on the nth day

In terms of these events,

a)

The probability of finding the dog if he searched in A is given by:

The probability of finding the dog if he searched in B is given by:

Therefore, Oscar should search in forest A.

b)

For this part, we assume that A and SA1 are independent events.

c)

Assume P(SA1) = 1/2

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