1. Binomial Distributions
Section 4.2
Binomial Distributions
Larson/Farber 4th ed 26

2. 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
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3. 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.
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4. 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.
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5. 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.
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6. 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

7. 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)
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8. 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

9. 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

10. Solution: Finding Binomial Probabilities
Method 1: Draw a tree diagram and use the Multiplication Rule 37

11. Solution: Finding Binomial Probabilities
Method 2: Binomial Probability Formula =
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12. 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

13. 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
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14. 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 ≈
P(x = 1) = 5C1(0.38)1(0.62)4 = 5(0.38)1(0.62)4 ≈
P(x = 2) = 5C2(0.38)2(0.62)3 = 10(0.38)2(0.62)3 ≈
P(x = 3) = 5C3(0.38)3(0.62)2 = 10(0.38)3(0.62)2 ≈
P(x = 4) = 5C4(0.38)4(0.62)1 = 5(0.38)4(0.62)1 ≈
P(x = 5) = 5C5(0.38)5(0.62)0 = 1(0.38)5(0.62)0 ≈ 41

15. Solution: Constructing a Binomial Distributionx
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 ≈ 1.
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16. 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).
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17. Solution: Finding Binomial Probabilities
P(x = 2) = 4C2(0.10)2(0.90)2 = 6(0.10)2(0.90)2 ≈
P(x = 3) = 4C3(0.10)3(0.90)1 = 4(0.10)3(0.90)1 ≈
P(x = 4) = 4C4(0.10)4(0.90)0 = 1(0.10)4(0.90)0 ≈
P(x ≥ 2) = P(2) + P(3) + P(4) ≈ ≈
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18. 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
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19. Example: Finding Binomial Probabilities Using a Table
# 26 on page 218 of the book x
Probability 1 2 3 4 5 x
Probability 1 2 3 4 5
Solution:
Binomial: n = 5, p = 0.25, q = 0.75, x = 0,1,2,3,4,5 47

20. Mean, Variance, and Standard Deviation
Mean: μ = np
Variance: σ2 = npq
Standard Deviation:
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21. 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

22. 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