Oscar的狗丢失在丛林中（条件与独立性概率习题）

题意如下

Oscar has lost his dog in either forest A (with probability 0.4) or in forest B (with probability 0.6).

If the dog is in forest A and Oscar spends a day searching for it in forest A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog is in forest B and Oscar spends a day looking for it there, he will find the dog that day with probability 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 overnight.

The dog is alive during day 0, when Oscar loses it, and during day 1, when Oscar starts searching. It is alive during day 2 with probability 23 . In general, for n≥1 , if the dog is alive during day n - 1, then the probability it is alive during day n is 2n+1 . The dog can only die overnight. Oscar stops searching as soon as he finds his dog, either alive or dead.

为了便于解题，先定义如下事件：

问题

a) In which forest should Oscar look on the first day of the search to maximize the probability he finds his dog that day?

b) Oscar looked in forest A on the first day but didn’t find his dog. What is the probability that the dog is in forest A?

c) Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day. What is the probability that he looked in forest A?

d) Oscar decides to look in forest A for the first two days. What is the probability that he finds his dog alive for the first time on the second day?

e) Oscar decides to look in forest A for the first two days. Given that he did not find his dog on the first day, find the probability that he does not find his dog dead on the second day.

f) Oscar finally finds his dog on the fourth day of the search. He looked in forest A for the first 3 days and in forest B on the fourth day. Given this information, what is the probability that he found his dog alive?