Larson/Farber 4th ed
Section 4.2
Binomial Distributions

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 p = P(S)
The number of times a trial is repeated
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 successes - favorite cookie is Oatmeal raisin.

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
Larson/Farber 4th ed
= 0.0486

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) =
5
C
0
(0.38) 0 (0.62) 5 = 1(0.38) 0 (0.62) 5 ≈ 0.0916
P ( x = 1) =
5
C
1
(0.38) 1 (0.62) 4 = 5(0.38) 1 (0.62) 4 ≈ 0.2807
P ( x = 2) =
5
C
2
(0.38) 2 (0.62) 3 = 10(0.38) 2 (0.62) 3 ≈ 0.3441
P ( x = 3) =
5
C
3
(0.38) 3 (0.62) 2 = 10(0.38) 3 (0.62) 2 ≈ 0.2109
P ( x = 4) =
5
C
4
(0.38) 4 (0.62) 1 = 5(0.38) 4 (0.62) 1 ≈ 0.0646
P ( x = 5) =
5
C
5
(0.38) 5 (0.62) 0 = 1(0.38) 5 (0.62) 0 ≈ 0.0079

1
2 x
0
3
4
5
Solution: Constructing a Binomial
Distribution
P(x)
0.0916
0.2808
0.3441
0.2109
0.0646
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

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) =
4
C
2
(0.10) 2 (0.90) 2 = 6(0.10) 2 (0.90) 2 ≈ 0.0486
P ( x = 3) =
4
C
3
(0.10) 3 (0.90) 1 = 4(0.10) 3 (0.90) 1 ≈ 0.0036
P ( x = 4) =
4
C
4
(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
2
0.395507813
0.263671875
3
4
5
0.087890625
0.014648438
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:
σ 2 = 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: σ 2 = 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