# Question

Bayes' theorem gives us a way of computing**P(A|B)**but uses

**P(B|A)**. How would we go about finding either without one being given initially?

# Answer

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

**A**and

**B**are true. We learned about

**P(A ∩ B)**when

**A**and

**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 **A** will happen if we already know that **B** happened?

Written like this makes it clear that we’re asking a question about both **A** and **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 **0** and **1** but instead between **0** and **P(B)**. Tying this all together, we can rewrite **P(A|B)** as follows

**P(A|B) = P(A ∩ B) / P(B)**

This presents us with another way of computing conditional probabilities: we can compute the probability of the intersection.