How to rigorously determine whether two events are independent?

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:

$A:$ Event that the pet is forest A.
$F_i:$ Event that he finds his pet on day $i.$
$S_i:$ Event that he searches forest A on day $i.$

The question asks which one of $ P(F_i | S_i) $ and $ P(F_i | S_i') $ 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 $S_i$ and $A$ 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. By common sense, I have concluded that $S_i$ and $A$ must be independent. When I solve the question using this assumption, I do the algebra and get $P(F_i | S_i) = 0.1 $ and $ P(F_i | S_i') = 0.09 $.

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) $. I want 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) .$

Answer 1

If Tim searches forest A the probability that he finds the pet that day is: $0.40\cdot 0.25$.   That is: $0.10$

If Tim searches forest B the probability that he finds the pet that day is: $0.60\cdot 0.15$.   That is: $0.09$

Tim searches in forest A first.

Answer 2

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) .$

There are exactly three possibilities:

• if knowing that $A$ happens doesn't change $B$'s probability, then $A$ and $B$ are independent;
• if $P(A)=0,$ then $A$ and $B$ are independent;
• otherwise, then $A$ and $B$ are not independent.

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.)

The definition that $A$ and $B$ being independent precisely means that $P(A)\:P(B)=0$ is the starting point that the bulleted schema/theorem above is derived from. You can think of the first bullet point as having motivated the definition.

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 $0.15.$ In which forest should Tim look on the first day of the search to maximize the probability he finds his pet that day?

$A:$ Event that the pet is forest A.
$F_i:$ Event that he finds his pet on day $i.$
$S_i:$ Event that he searches forest A on day $i.$

After, say, day $\mathit{\mathbf2},$ the probability experiment has sample space $$\{af_1,af_1'f_2,af_1'f_2',a'f_1,a'f_1'f_2,a'f_1'f_2'\}.$$ In this case: $$A=\{af_1,af_1'f_2,af_1'f_2'\}\\ F_1=\{af_1,a'f_1,\}\\ F_2=\{af_1'f_2,a'f_1'f_2\}.$$

As you can see, $\mathit{S_1}$ is not a subset of the sample space, so it cannot possibly be an event of our probability experiment!

The question asks which one of $ P(F_i | S_i) $ and $ P(F_i | S_i') $ is bigger. By common sense, I have concluded that $S_i$ and $A$ must be independent. When I solve the question using this assumption, I get $ P(F_i | S_i) = 0.1 $ and $ P(F_i | S_i') = 0.09 .$

No, the question asks whether $P(A\cap F_1)$ or $P(A'\cap F_1)$ is bigger.

$P(A\cap F_1)=0.4\times0.25=0.1,$ while $P(A'\cap F_1)=0.6\times0.15=0.09.$

Answer 3

The problem presents a narrative (i.e. non-mathematical) scenario. You used that narrative to create a Venn diagram. The process by which you converted the narrative to the diagram was not "rigorous," in the usual sense of the word. It is a process that involves a combination of intuition and mathematics. Once you have the Venn diagram (which is a representation of a probability space), you can do rigorous mathematics.

Now, the problem only asks you to consider the first day. So you only need $i = 1$. You cannot mathematically determine the independence of $S_1$ and $A$ from your Venn diagram. This should be clear. Simply find valid values of $x$ and $k$ that make $S_1$ and $A$ dependent.

This means you need to go back to the narrative and continue the process you started earlier. That is, you must use a combination of intuition and mathematics to argue that the independence of $S_1$ and $A$ is something that should be added to your Venn diagram.

Here is one possible line of argumentation: The event $S_1$ is a choice that Tim makes, prior to doing any searches whatsoever. Also, $P(A) = 0.4$ is a probability that is based on Tim's knowledge before doing any searches. Mathematics tells us that if $S_1$ and $A$ were not independent, then either $P(A \mid S_1) > P(A)$ or $P(A \mid S_1) < P(A)$. If that were the case, then somehow, the very act of choosing would change the probability of $A$. Since this doesn't make any sense, we should stipulate that they are independent.