Binomial Distributions 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 The experiment is repeated for a fixed number of trials, where each trial is independent of other trials. 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). The probability of a success P(S) is the same for each trial. 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. 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 Each question represents a trial. There are 12 adults questioned, and each one is independent of the others. There are only two possible outcomes of interest for the question: Oatmeal Raisin (S) or not Oatmeal Raisin (F). The probability of a success, P(S), is 0.10, for oatmeal raisin. The random variable x counts the number of successes - favorite cookie is Oatmeal raisin. 31

Solution: Binomial Experiments 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 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 List the possible values of x with the corresponding probability of each. Example: Binomial probability distribution for Oatmeal Cookies: n = 12 , p = 0.10 Use binomial probability formula to find probabilities. x 1 2 3 ... P(x) 0.283 0.377 0.230 0.085 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) 0.0916 1 0.2808 2 0.3441 3 0.2109 4 0.0646 5 0.0079 0.9999 All of the probabilities are between 0 and 1 and the sum of the probabilities is 0.9999 ≈ 1. 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 1 2 3 4 5 x Probability 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 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