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Section 4.2 Binomial Distributions Larson/Farber 4th ed 26 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 27 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 28 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 29 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 30 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 31 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 32 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 35 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 36 Solution: Finding Binomial Probabilities Method 1: Draw a tree diagram and use the Multiplication Rule 37 Solution: Finding Binomial Probabilities Method 2: Binomial Probability Formula = 0.0486 Larson/Farber 4th ed 38 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 39 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 40 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 41 Solution: Constructing a Binomial Distribution x P(x) 0 0.0916 All of the probabilities are between 0 and 1 and the sum of the 1 0.2808 probabilities is 0.9999 ≈ 1. 2 0.3441 3 0.2109 4 0.0646 5 0.0079 0.9999 Larson/Farber 4th ed 42 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. Solution: • n = 4, p = 0.10, q = 0.90 • At least two means 2 or more. • Find the sum of P(2), P(3) and P(4). Larson/Farber 4th ed 43 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 44 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 45 Example: Finding Binomial Probabilities Using a Table # 26 on page 218 of the book x Probability x Probability 0 0 0.237304688 1 1 0.395507813 2 2 0.263671875 3 3 0.087890625 4 4 0.014648438 5 5 0.000976563 Solution: • Binomial: n = 5, p = 0.25, q = 0.75, x = 0,1,2,3,4,5 47 Mean, Variance, and Standard Deviation • Mean: μ = np • Variance: σ2 = npq • Standard Deviation: Larson/Farber 4th ed 51 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: 52 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 54

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