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Section 4.2 Binomial Distributions Larson/Farber 4th ed Section 4.2 Objectives • Determine if a probability experiment is a binomial experiment • Find binomial probabilities using the binomial probability formula • Find binomial probabilities using technology and a binomial table • Graph a binomial distribution • Find the mean, variance, and standard deviation of a binomial probability distribution Larson/Farber 4th ed Binomial Experiments 1. The experiment is repeated for a fixed number of trials, where each trial is independent of other trials. 2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F). 3. The probability of a success P(S) is the same for each trial. 4. The random variable x counts the number of successful trials. Larson/Farber 4th ed Notation for Binomial Experiments Symbol Description n The number of times a trial is repeated p = P(S) The probability of success in a single trial q = P(F) The probability of failure in a single trial (q = 1 – p) x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, … , n. Larson/Farber 4th ed Example: Binomial Experiments Decide whether the experiment is a binomial experiment. If it is, specify the values of n, p, and q, and list the possible values of the random variable x. 1. Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 12 adults and ask each to name his or her favorite cookie. Larson/Farber 4th ed Solution: Binomial Experiments Binomial Experiment 1. Each question represents a trial. There are 12 adults questioned, and each one is independent of the others. 2. There are only two possible outcomes of interest for the question: Oatmeal Raisin (S) or not Oatmeal Raisin (F). 3. The probability of a success, P(S), is 0.10, for oatmeal raisin. 4. The random variable x counts the number of Solution: Binomial Experiments Binomial Experiment • n = 12 (number of trials) • p = 0.10 (probability of success) • q = 1 – p = 1 – 0.10 = 0.90 (probability of failure) • x = 0, 1, 2, 3, 4, 5, 6, 7, 8 (number of people that like oatmeal raisin cookies) Larson/Farber 4th ed Binomial Probability Formula Binomial Probability Formula • The probability of exactly x successes in n trials is • • • • n = number of trials p = probability of success q = 1 – p probability of failure x = number of successes in n trials Larson/Farber 4th ed Example: Finding Binomial Probabilities Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 4 adults and ask each to name his or her favorite cookie. Find the probability that the number who say oatmeal raisin is their favorite cookie is (a) exactly 2, (b) at least 1 and (c) less than four Larson/Farber 4th ed Solution: Finding Binomial Probabilities Method 1: Draw a tree diagram and use the Multiplication Rule Solution: Finding Binomial Probabilities Method 2: Binomial Probability Formula = 0.0486 Larson/Farber 4th ed Binomial Probability Distribution Binomial Probability Distribution • List the possible values of x with the corresponding probability of each. • Example: Binomial probability distribution for Oatmeal Cookies: n = 12 , p = 0.10 x 0 1 2 3 ... P(x) 0.283 0.377 0.230 0.085 ... Use binomial probability formula to find probabilities. Larson/Farber 4th ed Example: Constructing a Binomial Distribution Thirty eight percent of people in the United States have type O+ blood. You randomly select five Americans and ask them if their blood type is O+. •Construct a binomial distribution Larson/Farber 4th ed Solution: Constructing a Binomial Distribution • 38% of Americans have blood type O+. • n = 5, p = 0.38, q = 0.62, x = 0, 1, 2, 3, 4, 5 P(x = 0) = 5C0(0.38)0(0.62)5 = 1(0.38)0(0.62)5 ≈ 0.0916 P(x = 1) = 5C1(0.38)1(0.62)4 = 5(0.38)1(0.62)4 ≈ 0.2807 P(x = 2) = 5C2(0.38)2(0.62)3 = 10(0.38)2(0.62)3 ≈ 0.3441 P(x = 3) = 5C3(0.38)3(0.62)2 = 10(0.38)3(0.62)2 ≈ 0.2109 P(x = 4) = 5C4(0.38)4(0.62)1 = 5(0.38)4(0.62)1 ≈ 0.0646 P(x = 5) = 5C5(0.38)5(0.62)0 = 1(0.38)5(0.62)0 ≈ 0.0079 Solution: Constructing a Binomial Distribution x P(x) 0 0.0916 1 0.2808 2 0.3441 3 0.2109 4 0.0646 5 0.0079 0.9999 Larson/Farber 4th ed All of the probabilities are between 0 and 1 and the sum of the probabilities is 0.9999 ≈ 1. Example: Finding Binomial Probabilities Ten percent of adults say oatmeal raisin is their favorite cookie. You randomly select 4 adults and ask each if their favorite cookie is oatmeal raisin. Find the probability that at least two answer that their favorite cookie is oatmeal raisin. Solution: • n = 4, p = 0.10, q = 0.90 • At least two means 2,3 and 4. • Find the sum of P(2), P(3) and P(4). Larson/Farber 4th ed Solution: Finding Binomial Probabilities P(x = 2) = 4C2(0.10)2(0.90)2 = 6(0.10)2(0.90)2 ≈ 0.0486 P(x = 3) = 4C3(0.10)3(0.90)1 = 4(0.10)3(0.90)1 ≈ 0.0036 P(x = 4) = 4C4(0.10)4(0.90)0 = 1(0.10)4(0.90)0 ≈ 0.0001 P(x ≥ 2) = P(2) + P(3) + P(4) ≈ 0.0486 + 0.0036 + 0.0001 ≈ 0.0523 Larson/Farber 4th ed Example: Finding Binomial Probabilities Using Technology Thirty eight percent of people in the United States have type O+ blood. You randomly select 138 Americans and ask them if their blood type is O+. What is the probability that exactly 23 have blood type O+? Solution: • Binomial with n = 138, p = 0.38, q=0.62, x = 23 Larson/Farber 4th ed Example: Finding Binomial Probabilities Using a Table # 26 on page 218 of the book x Probability 0 1 2 3 4 5 x Probability 0 0.237304688 1 0.395507813 2 0.263671875 3 0.087890625 4 0.014648438 5 0.000976563 Solution: • Binomial: n = 5, p = 0.25, q = 0.75, x = 0,1,2,3,4,5 Mean, Variance, and Standard Deviation • Mean: μ = np • Variance: σ2 = npq • Standard Deviation: Larson/Farber 4th ed Example: Finding the Mean, Variance, and Standard Deviation Fourteen percent of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each if cashews are their favorite nut. Find the mean, variance and standard deviation. Solution: n = 12, p = 0.14, q = 0.86 Mean: μ = np = (12)∙(0.14) = 1.68 Variance: σ2 = npq = (12)∙(0.14)∙(0.86) ≈ 1.45 Standard Deviation: Section 4.2 Summary • Determined if a probability experiment is a binomial experiment • Found binomial probabilities using the binomial probability formula • Found binomial probabilities using technology and a binomial table • Graphed a binomial distribution • Found the mean, variance, and standard deviation of a binomial probability distribution Larson/Farber 4th ed

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