Bayes’ theorem provides a way of computing
P(A|B) but uses
P(B|A). How would we go about finding either without one being given initially?
To think about this question more clearly, we need to think about the intersection: P(A ∩ B). Recall that P(A ∩ B) is the probability that both
B are true. We learned about P(A ∩ B) when
B were independent but it also makes sense to think about the intersection in other cases.
First, let’s rephrase the conditional probability in plain language.
P(A|B) is asking:
What is the probability that
Awill happen if we already know that
Written like this makes it clear that we’re asking a question about both
B happening. So
P(A|B) is related to P(A ∩ B) in some way but how exactly? There certainly not equal. This is where our knowledge that
B already happened comes into play. Since
B happened, the probability that we’re computing is essentially no longer between
1 but instead between
P(B). Tying this all together, we can rewrite
P(A|B) as follows
P(A|B) = P(A ∩ B) /
This presents us with another way of computing conditional probabilities: we can compute the probability of the intersection.