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The Binomial Distribution Section 15.8 A binomial distribution is a discrete distribution defined by two parameters: The number of trials, n The probability of success, p The discrete random variable (X) is applied to sampling with replacement. The probability distribution associated with this variable is the binomial probability distribution. (For sampling without replacement, the hypergeometric probability distribution is the model used but is not a part of this course!) X~B ( n, p) is written to indicate that the discrete random variable X is binomially distributed. “ ~ “ reads “is distributed as” Properties of a Binomial Experiment There is a fixed number of trials: n trials On each one of the n trials, there is only one of two possible outcomes, labeled “success” or “failure” (Bernoulli Trials) Each trial is identical and independent On each of the trials, the probability of a success , p, is always the same, and the probability of a failure, q = 1 – p, is also always the same. The binomial probability distribution function n r nr p (1 p) r For example, the probability function for X~B(6, o.4) means that there are 6 trials and the probability of success is 0.4. 6 x 6 x P( X x) 0.4 0.6 x If X~B(5, 0.6), find P(X=4) Example Suppose a spinner has 3 blue edges and 1 white edge. For each spin we will get either a blue or a white edge. If we call a blue result a “success” and a white result a “failure”, then we have a binomial experiment. Let the random variable X be the number of “successes” or blue results. P(success)= P(failure)= Consider twirling the spinner 3 times, n = 3. Therefore, X = 0, 1, 2, or 3.