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## Independence

Suppose that we were looking at coffee drinking and CHD, and we found that the probability of CHD in the study population was 0.1. Then suppose we knew that the probability of being a coffee drinker was 0.3, and that the probability of having CHD and being a coffee drinker was 0.03. Then we have 3 percent of the population having both CHD and coffee drinking behavior, and 30% of the population being coffee drinkers. Then the conditional probability of having CHD given that a person is a coffee drinker is 0.03/0.3=0.1. Knowing that a person was a coffee drinker did not change your estimate of the probability of having CHD.
In general, whenever the conditional probability of A given B is the same as the probability of A, the events A and B are said to be independent. Since (when P(B) isn't zero) conditional probability is defined according to P(A|B)=P(A B)/P(B), if we substitute P(A) for P(A|B), we find that when the events A and B are independent, that P(A) = P(A B)/P(B). But we can multiply both sides by P(B), and we find that P(A B) = P(A) P(B). In other words, when A and B are independent, then the probability of A and B happening is the probability that A happens times the probability that B happens.

On to the binomial distribution.