Chapter 4

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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

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

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
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
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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