If we want to know the probability of two events happening, we can say
P(A and B) = P(A)P(B)
At least, that is what we are taught in intro to statistics.
This only works if A and B are not relevant to each other, and that knowing A does not affect anything about B. Not really useful when we are trying to research the effects of one factor (A) over another factor (B).
Thus, we can adjust the equation to say
P(A)P(B|A) = if both A and B happen, A can be true, can I figure out B knowing A is true?
Lets say that you want to know if I am married.
You might first look for a ring, (A), and then try to find the probability of whether or not I am married (B) based on whether or not I have a ring (A). (Is the chance of me being married higher if I have a ring?)
I can switch it around too.
P(B)P(A|B) = if you know that I am married, does this affect the probability if I have a ring?
The funny thing is, these two combine into
P(A and B) = P(A)P(B|A) = P(B)P(A|B)
P(A|B)= the conditional probability of A, given B,
= P(A)P(B|A) / P(B) = Bayes’ Theorem