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Probabilities of Complementary Events

     Remember that if A is some event, and Ac is the complement of A in the sample space , then A and Ac are disjoint (mutually exclusive, have nothing in common). Also, their union is the entire sample space: A Ac = . Since is the same set as A Ac, it has to have the same probability. Moreover, we know that probability has to be 1: 1=P() = P(A Ac). But that is a disjoint union, and the probability of the union is the sum of the probabilities: 1=P(A)+P(Ac). So what this amounts to is saying that the probability that something happens, plus the probability that it doesn't happen, is one. So for instance if you know 20% of your sample are smokers, then you know 80% have to be nonsmokers.

Probability of Empty Event is zero

     Since the empty set (impossible event) has nothing in common with anything, we can write the sample space as the union of the sample space with the empty set. Then the probability of the disjoint union is the sum of the probabilies, so P( ) = P( ) = P( ) + P( ) So we get 1=1+P( ) . Subtract off one from either side, and we learn that the probability of something impossible has to be zero.
On to the Bernoulli distribution.

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