This equation defines the probability of event B given event A.

For example, if event A is the probability of picking a student with a red shirt and event B is the probability of picking a student with jeans, then the probability of B given A is the probability of picking a student with jeans out of all the students wearing red shirts.

You must know the probability of B and A (the probability that both A and B being true) to use this equation.

author: Koby Chan

Notes

In our example of a set of student you might count the number of student wearing both jeans and red shirts.  This would be the probability of both occurring P(A&B).  Let's say we have 150 students in an auditorium and we ask anyone wearing both jeans and a red shirt to raise their hands.  We count 22 hands.  So, P(A&B) = 22/150 = 0.147

Now, we just have those who are wearing red shirts raise their hands.  The number of students wearing red shirts must be equal to or greater than the number of students red shirts who are wearing red shirts (independent of what kind of pants they wear).

Let's say 25 are wearing a red shirt.  So, P(B) = 25/150 = 0.167

We don't have to have the students raise their hands to know the probability of the picking a student from those wearing red shirts that also is wear wearing jeans:

P(A) = P(A&B) / P(B)

P(A) = 0.147 / 0.167 = 0.88

In our example there is a 88% change we would pick from those wearing red shirts and randomly also get a student wearing jeans.