Binomial Distributions 4.2 SECTION 4.2 Binomial Distributions 201 Binomial Experiments Binomial Probability Formula Finding Binomial Probabilities Graphing Binomial Distributions Mean, Variance, and Standard Deviation What You Should Learn How to determine whether a probability experiment is a binomial experiment How to find binomial probabilities using the binomial probability formula How to find binomial probabilities using technology, formulas, and a binomial probability table How to construct and graph a binomial distribution How to find the mean, variance, and standard deviation of a binomial probability distribution Binomial Experiments There are many probability experiments for which the results of each trial can be reduced to two outcomes: success and failure. For instance, when a basketball player attempts a free throw, the player either makes the basket or does not. Probability experiments such as these are called binomial experiments. A binomial experiment is a probability experiment that satisfies these conditions. 1. The experiment has a fixed number of trials, where each trial is independent of the other trials. 2. There are only two possible outcomes of interest for each trial. Each outcome can be classified as a success (S) or as a failure (F). 3. The probability of a success is the same for each trial. 4. The random variable x counts the number of successful trials. DEFINITION Symbol Description n The number of trials p The probability of success in a single trial q The probability of failure in a single trial 1q = 1 - p2 x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, . . ., n. Notation for Binomial Experiments In a binomial experiment, success does not imply something good occurred. For instance, in an experiment a survey asks 1012 people about identity theft. A success is a person who was a victim of identity theft. Here is an example of a binomial experiment. From a standard deck of cards, you pick a card, note whether it is a club or not, and replace the card. You repeat the experiment five times, so n = 5. The outcomes of each trial can be classified in two categories: S = selecting a club and F = selecting another suit. The probabilities of success and failure are p = 1 4 and q = 1 - 1 4 = 3 4 . The random variable x represents the number of clubs selected in the five trials. So, the possible values of the random variable are x = 0, 1, 2, 3, 4, 5. For instance, if x = 2, then exactly two of the five cards are clubs and the other three are not clubs. An example of an experiment with x = 2 is shown at the left. Note that x is a discrete random variable because its possible values can be counted. 1 Trial F 2 S 3 F 4 F 5 S Outcome S or F? There are two successful outcomes. So, x = 2.
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