5 Random Variables Lesson 5 3 Binomial Random
5 Random Variables Lesson 5. 3 Binomial Random Variables Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers
Binomial Random Variables Learning Targets After this lesson, you should be able to: ü Determine whether or not a given scenario is a binomial setting. ü Calculate probabilities involving a single value of a binomial random variable. ü Make a histogram to display a binomial distribution and describe its shape. Statistics and Probability with Applications, 3 rd Edition 2
Binomial Random Variables When the same chance process is repeated several times, we are often interested in how many times a particular outcome occurs. Binomial Setting A binomial setting arises when we perform several independent trials of the same chance process and record the number of times that a particular outcome (called a “success”) occurs. B I N S The four conditions for a binomial setting are: • Binary? The possible outcomes of each trial can be classified as “success” or “failure. ” • Independent? Trials must be independent; that is, knowing the result of one trial must not tell us anything about the result of any other trial. • Number? The number of trials n of the chance process must be fixed in advance. • Success? There is the same probability p of success on each trial. Statistics and Probability with Applications, 3 rd Edition 3
Binomial Random Variables When checking the Binary condition, note that there can be more than two possible outcomes per trial. In the Pop Quiz activity, each question (trial) had five possible answer choices: A, B, C, D, or E. If we define “success” as guessing the correct answer to a question, then “failure” occurs when the student guesses any of the four incorrect answer choices. Binomial Random Variable The count of successes X in a binomial setting is a binomial random variable. The possible values of X are 0, 1, 2, . . . , n. Statistics and Probability with Applications, 3 rd Edition 4
Binomial Random Variables How can we calculate probabilities involving binomial random variables? Binomial Probability Formula Suppose that X is a binomial random variable with n trials and probability p of success on each trial. The probability of getting exactly k successes in n trials (k = 0, 1, 2, . . . , n) is where The binomial probability formula looks complicated, but it is fairly straight forward if you know what each part of the formula represents: P(X = k) = (# of ways to get k successes in n trials) • (success probability)k(failure probability)n−k Statistics and Probability with Applications, 3 rd Edition 5
Binomial Random Variables What does the probability distribution of a binomial random variable look like? Binomial Distribution The probability distribution of a binomial random variable is a binomial distribution. Any binomial distribution is completely specified by two numbers: the number of trials n of the chance process and the probability p of success on any trial. X = the number of children with type O blood This binomial distribution with n = 5 and p = 0. 25 has a clear right-skewed shape. Statistics and Probability with Applications, 3 rd Edition 6
LESSON APP 5. 3 Is the train binomial? According to New Jersey Transit, the 8: 00 A. M. weekday train from Princeton to New York City has a 90% chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let X = the number of days on which the train arrives on time. 1. Explain why this is a binomial setting. 2. Calculate and interpret P(X = 4). 3. Make a histogram of the probability distribution of X. Describe its shape. Statistics and Probability with Applications, 3 rd Edition 7
Binomial Random Variables Learning Targets After this lesson, you should be able to: ü Determine whether or not a given scenario is a binomial setting. ü Calculate probabilities involving a single value of a binomial random variable. ü Make a histogram to display a binomial distribution and describe its shape. Statistics and Probability with Applications, 3 rd Edition 8
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