Topic: Independent Events

Related Topics Probability Measure Theory Random Variables

The Multiplication Rule for Independent Events states that if two events, A and B, are independent, then P(A B) = P(A) * P(B). If n events, A1, A2, ..., An, are independent, then P(A1 A2 ... An) = P(A1) * P(A2) * ... * P(An). This probability law is also called the product rule. The rule simply states that the probability of the joint occurrence of mutually independent events is the product of their probabilities. When flipping a coin, each flip is an independent event. This means that the results of each flip are in no way influenced by the results of previous flips. Try flipping the coin below several times, and get a sense for how often a "head" will be followed by another "head". In the experiment below, coins are laid end-to-end whenever they match the previous flip. Run this experiment for several hundred flips, to get a sense of the likelihood of getting long runs of all heads or all tails. The illustration below shows the results of one experiment of 16,151 runs. Note the pattern in the frequency of runs. This experimental data supports the Multiplication Rule for Independent Events, since the actual number of occurrences of a run of length i is approximately (1 / 2) ^ i.