I have a few things I am hoping to get clarified. Let's say we want to determine $Pr(A \cap B)$.
- Can we say $Pr(A \cap B)$ is the joint probability of A and B occurring? Or is that the wrong term to use? In calculating conditional probabilities, this seems like it's the correct term to use to describe the numerator. Assuming A and B are independent, we have $Pr(A \cap B) = Pr(A) * Pr(B)$. If dependent, we have $Pr(A \cap B) = Pr(A | B) * Pr(B)$
- However, we also have the formula $Pr(A \cap B) = Pr(A) + Pr(B) - Pr(A \cup B)$. In this case, would you still describe $Pr(A \cap B)$ as a joint probability?
If you were explaining the above concept to someone completely new at statistics, how would you explain the situation where we have two formulas for $Pr(A \cap B)$? To be more specific, how would you explain to someone when to use which formula and why there are two formulas?