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When considering mutually exclusive events A and B (think sides of a die), the probability of A or B occurring is the sum of their individual probabilities - P(A or B) = P(A) + P(B).
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P(A|B) = P(A,B)/P(B): A conditional probability is the probability that both A and B occur, divided by the probability that B occurs. Another way to think about this is as the fraction of B events that also include A. Yet another way to think about this is to ask, what have we learned about the probability of A, given that we know B occurred?
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If A and B are independent, P(A,B) = P(A)P(B). The probability that two events occur is the product of their individual probabilities. Similarly, P(A|B) = P(A). If they're independent, we don't learn anything about the probability of A given that we know B occurred.