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Class Lecture

# Class Lecture - Statistical Methods Binomial Hypergeometric...

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Statistical Methods Binomial & Hypergeometric Probability Distributions

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Objectives By the end of this lesson, students should be able to: Identify an experiment as binomial by using the four requirements of a binomial experiment. Calculate the mean and standard deviation of a binomial probability distribution. Compute binomial probabilities. Identify an experiment as hypergeometric. Compute the mean and standard deviation of a hypergeometric distribution. Compute hypergeometric probabilities. Introduction In probability and statistics there are two important and common discrete probability distributions that need be discussed. These are the binomial probability distribution and the hypergeometric probability distribution. The binomial probability distribution consists of independent trials and constant probabilities, such as those that arise from sampling with replacement or sampling without replacement from a relatively infinite population. The hypergeometric probability distribution consists of dependent trials and conditional probabilities, such as those that arise from sampling without replacement from a relatively small population. Binomial Probability Distributions A Bernoulli trial is a trial that has only two possible outcomes, success and failure. A series of n independent Bernoulli trials in which the probability of success remains constant comprises a binomial experiment . We denote the probability of success in each trial as p so that the probability of failure is 1 – p . Note that 1 – p + p = 1 which is the probability that something in the sample space occurs. The only two elements in the sample space are success and failure so the sum of their probabilities must be 1. The Four Requirements for a Binomial Experiment: 1. There are a fixed number of trials, n . 2. There are two possible outcomes per trial: success and failure. 3. The trials are independent of each other. 4. The probabilities remain constant from trial to trial: P(success) = and P(failure) = 1 . Binomial & Hypergeometric Probability Distributions Page 2 of 10
When calculating the probability of x successes in a binomial experiment, it is first necessary to determine the number of ways to get x successes in n trials, or the number of combinations of x items from n available items. The number of combinations is denoted as n C x , read “ n choose x .” In general, the number of ways to choose (order irrelevant) r objects from n distinguishable objects is , where represents the number of permutations/arrangements of r elements out of n distinguishable elements and r ! is the number of ways to arrange/permute r elements. Both n P r and n C r are programmed into most scientific calculators. These functions are also available in EXCEL as PERMUT and COMBIN.

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Class Lecture - Statistical Methods Binomial Hypergeometric...

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