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Handout_6a THE BINOMIAL DISTRIBUTION

# Handout_6a THE BINOMIAL DISTRIBUTION - May 1109 THE...

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May 11’09 THE BINOMIAL PROBABILITY DISTRIBUTION #6a Binomial Probability Experiment is an experiment that is made up of repeated trials of the same basic experimental event. Such an experiment must possess the following properties: 1. Each trial has only two possible outcomes (usually labeled as a success or a failure ). 2. There are n repeated independent trials, so that the probability of success remains the same for each trial. 3. The number x of successful trials may take on any integer value from 0 to n , and may serve as the binomial random variable . If the probability of success on each individual trial is p and the probability of failure is 1 – p = q , then the binomial probability P(x) of exactly x successes and (n – x) failures in n independent trials is P(x) = n C x p x q n – x = n! p x q n – x /[x!(n – x)!] (6a.1) In (6a.1) n C x is a combination of n elements taken r at a time (see Handout_6 …) The expressions P(x) (6a.1) are the terms in the expansion of the binomial (see Handout #6) (p + q) n = 1 n = 1 = Σ P(x) = p n + n C 1 *p n–1 q + n C 2 *p n–2 q 2 + … + n C m *p n–m q m + …+ q n all x NOTE: “Independent” trials mean that the result of one trial does not affect the probability of success of any other trial in the experiment. In other words, the probability of “success” remains constant throughout the entire experiment.

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