Consider the following question:
Tim has lost his pet in either forest A (with probability 0.4) or in forest B (with probability 0.6).
If his pet is in forest A and Tim spends a day searching for it in forest A, the conditional probability that he will find his pet that day is 0.25. Similarly, if his pet is in forest B and Tim spends a day looking for it there, he will find the pet that day with probability 0.15.
The pet cannot go from one forest to the other. Tim can search only in the daytime, and he can travel from one forest to the other only overnight. Tim stops searching as soon as he finds his pet.
In which forest should Tim look on the first day of the search to maximize the probability he finds his pet that day?
To solve this question, I have defined the following events:
F: Event that the pet is forest A. F': Event that the pet is in forest B.
Fi: Event that he finds his pet on day i. Fi': Event that he doesn't find his pet on day i.
Si: Event that he searches forest A on day i. Si': Event that he searches forest B on day i.
And from the given information, I have constructed the following Venn diagram:
The question asks which one of $ P(Fi | Si) $ or $ P(Fi | Si') $ is bigger.
Now as far as I understand, solving the question with only this much information is not possible. I guess that in order to solve the question, I need to assume that Si and F are independent events. Intuitively, this is a sound assumption because by the wording of the question, it seems like Tim chooses the forest to search totally at his own wish.
When I solve the question using this assumption, I do the algebra and get $ P(Fi | Si) = 0.1 $ and $ P(Fi | Si') = 0.09 $.
Now comes my question: I have assumed that Si and F are independent totally intuitively. That is: I have read the question and by the wording of the question and by common sense, I have concluded that Si and F must be independent and hence, I was able to solve the question.
However, I am not exactly sure whether these two events are indeed independent. By common sense and intuition, I have concluded that these events are independent. However, I don't have mathematical arguments, or any kind of argument that is more rigorous than common sense and intuition to back up this assumption.
So my question is this: How can I rigorously determine whether two events are independent?
By the way, note that an argument such as: "To rigorously determine whether two events A and B are independent, simply check whether the equation $ P(A \cap B) = P(A) * P(B) $ holds." is totally invalid. Because I want to know whether events A and B are independent in the case that I don't know $ P(A \cap B) $ or $ P(A) $ or $ P(B) $. In other words, I want to know whether two events are independent in order to make inferences on $ P(A \cap B) $ or $ P(A) $ or $ P(B) $. Not to validate my calculations on $ P(A \cap B) $ and $ P(A) $ and $ P(B) $.