What's the use of the conditional probability formula?

Question

First we have that the probability of $A$ occurring given that $B$ occurs is: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ but if the events are independent then that would simply equal $P(A)$ so what's the use of this formula? where do we need it? And I can't seem to make sense of the whole formula if the events are dependent so I can't really understand it. (professor didn't mention anything about what type of events this is used with)

Answer 1

You are correct, if $A$ and $B$ are independent, then $P(A\cap B) = P(A)P(B)$ and the entire expression is not very enlightening.

But when $A$ and $B$ are dependent, this result becomes very meaningful. For example, consider this problem: With the probability of $1/3$, exactly one of eight identical-looking envelopes contains a bill (conditional probability question) or the very famous Monty Hall puzzle.

Answer 2

You can use the formula whenever you want. If the events are independent you will have

$$\mathbb{P}[A|B]=\frac{\mathbb{P}[A]\times \mathbb{P}[B]}{\mathbb{P}[B]}=\mathbb{P}[A]$$

So you used the same formula...but with a simplified result