Discrete Probability Distributions Chapter 6 Lecture 11 Mc

Binomial Probability Distribution 6 -2 l A Widely occurring discrete probability distribution l The binomial random variable x is the count of the number of successful trials that occur; x may take on any integer value from zero to n

Characteristics of Binomial Probability Experiment The number of trials is fixed i. e. n For example: Ø 15 tosses of coin Ø 20 patients Ø 100 people surveyed 1. 6 -3

Characteristics of Binomial Probability Experiment 2. There are only two possible outcomes on a particular trial of an experiment. Ø An outcome on each trial of an experiment is classified into one of two mutually exclusive categories—a success or a failure. Examples: - The probability that 30 out of 40 statistics students will pass the final exam. - The probability that 5 out of 12 tosses of a coin will result in heads. 6 -4

3. Ø Ø Ø 6 -5 The probability of success and failure stay the same for each trial. Probability of success = or p Probability of failure = 1 - or q The probability you will guess the first question of a true/false test correctly (success) is ½. The probability that you will guess correctly on the second question is also ½, and so on.

4. Each trial is independent of any other trial l The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial. 6 -6

Binomial Probability Formula 6 -7

Binomial Probability - Example There are five flights daily from Pittsburgh via US Airways into the Bradford, Pennsylvania, Regional Airport. Suppose the probability that any flight arrives late is. 20. What is the probability that none of the flights are late today? 6 -8

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Example l. A coin is tossed 6 times. The probability of heads on any toss is 0. 3. Let X denote the number of heads that come up. Calculate P (X = 2) and P (X = 3). 6 -10

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Binomial Dist. – Mean and Variance 6 -12

Example For the example regarding the number of late flights, recall that =. 20 and n = 5. What is the average number of late flights? μ= n What is the variance of the number of late flights? 6 -13

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Binomial Dist. – Mean and Variance: Another Solution 6 -15

Binomial Probability Distributions Example A study by the Department of Transportation concluded that 76. 2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected. What is the probability the front seat occupants in exactly 7 of the 12 vehicles are wearing seat belts? 6 -16

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Cumulative Binomial Probability Distributions - Example A study by the Illinois Department of Transportation concluded that 76. 2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected. What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts? 6 -18

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Example l a. b. c. d. 6 -21 In a recent study 90 % of the homes in US were found to have large screen TVs. In a sample of 9 homes, what is the probability that: All nine have large screen TVs. Less than 5 have large screen TVs. More than 5 have large screen TVs. At least 7 homes have large screen TVs.