Conditional probability: why is it defined like that?

So, you want to calculate the probability of an event knowing that another has happened. There is a formula for that, it is called conditional probability, but why is it the way it is? Let’s first write down the definition of conditional probability:

    \[\mathbb{P}(A | B) = \dfrac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}\]

We need to wonder: what does the happening of event B tell about the odds of happening of event A? How much more likely A becomes if B happens? Think in terms of how B affects A.

If A and B are independent, then knowing something about B will not tell us anything at all about A, at least not that we did not know already. In this case A \cap B is empty and thus \mathbb{P}(A | B) = \mathbb{P}(A). This makes sense! In fact, consider this example: how does me buying a copybook affects the likelihood that your grandma is going to buy a frying pan? It does not: the first event has no influence on the second, thus the conditional probability is just the same as the normal probability of the first event.

Sets no intersection

If A and B are not independent, several things can happen, and that is where things get interesting. We know that B happened, and we should now think as if B was our whole universe. The idea is: we already know what are the odds of A, right? It is just \mathbb{P}(A). But how do they increase if we know that we do not really have to consider all possible events, but just a subset of them? As an example, think of \mathbb{P}(\text{drawing a red ball}) versus \mathbb{P}(\text{drawing a red ball}) knowing that all balls are red. This makes a huge difference, right? (As an aside, that is what we mean when we say that probability is a measure of our ignorance.)

So anyway, now we ask: what is the probability of A? Well, it would just be \mathbb{P}(A), but we must account for the fact that we now live inside B, and everything that is outside it is as if it did not existed. So \mathbb{P}(A) actually becomes \mathbb{P}(A \cap B): we only care about the part of A that is inside B, because that is where we live now.

But, there is a caveat. Continue reading “Conditional probability: why is it defined like that?”