Essentially, in conditional probability, when it says "given A" it is telling you that we know for sure that event A has occurred, so we are now only interested in outcomes that belong to A. That becomes the "whole". P(B|A) wants the fraction of that "whole" that also belongs to B.
For example, if you look at my
question 18 in the probability handout from my Basic Stats 1 Probability Lesson above, I could add a
part (d) that asks, "
What is the probability someone is a basketball fan if they are a hockey fan?" Any probability question that asks, what is the probability of B
if event A has occurred, you are doing conditional probability.
We want P(B|H). I first look through my Venn diagram and find all the bits that belong to H, since we know for sure the person is a hockey fan. There are four bits in the H circle so I add those bits up: 33 + 31 + 8 + 5 = 77%. Now, I gather all the bits in that H circle that represent people who are also basketball fans. There are two bits: 8 + 5 = 13%. Thus, the probability a person is a basketball fan if they are a hockey fan is 13%/77% or .13/.77 = .1688.
Here is a couple of extra conditional probability questions I have added to
question 4 in my Probability Lesson handout above:
Here is a couple of extra conditional probability questions I have added to
question 16 in my Probability Lesson handout above: