Two events are independent of one another if the occurance of one does not affect the (probability of) occurance of the other.
Let’s say we have two events, A and B. If they are independent, the probability of B is simply P(B), i.e. it doesn’t depend on A at all. The probability of both A and B, P(A and B), is P(A) * P(B).
If the two events are not independent, meaning the occurance of one either raises or lowers the probability of the other, then the probability of both of them occuring is
Which is the same as saying
Both of the above equations are saying the same thing, from slightly different perspectives. I.e. they are equivalent. If we set these equations equal to one another, then algebraically manipulate a little, we get Bayes’ formula
So, bayes formula is useful when two events are somehow related (i.e. one occuring changes the probability of the other occuring).
|So, if you know the probability of one given the other (e.g. $$ P(B||A) P(A||B) $$)! All you need to know is the probability of each individually.|
- add 2 examples (one extremely simple one, one “medical diagnosis” one)