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## The Bernoulli Distribution

Suppose you want to describe the probability that a person could become infected with Hepatitis C following a needlestick injury. The elementary events or outcomes are that the person became infected, or that the person did not. If you knew the probability, say p, of infection, then if a very large number of individuals are so exposed, then the frequency of infection ought to be close to p (the Law of Large Numbers).
More generally, you could have any situation with two outcomes, which could be 0 or 1, or perhaps live or dead, or success and failure. It's conventional to use the terms "success" and "failure"; we'll denote these by 1 and 0 respectively. The sample space = {0,1}.
The possible events are , {0}, {1}, and {0,1}. Each of these is a subset of the sample space, and the Bernoulli distribution assigns a probability to each of them:
 P() = 0 P({0}) = 1-p P({1}) = p P({0,1}) = 1
Actually, the first and last of these have to be true no matter what. And once you know that the probability of success is p (line three), you automatically know that P({0})=1-p.

On to conditional probability.