Download Section 4.2 Binomial Distributions Larson/Farber 4th ed

Section 4.2
Binomial Distributions
Larson/Farber 4th ed
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
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
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
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
= 0.0486
Larson/Farber 4th ed
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) = 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
Solution: Constructing a Binomial
Distribution
x
P(x)
0
0.0916
1
0.2808
2
0.3441
3
0.2109
4
0.0646
5
0.0079
0.9999
Larson/Farber 4th ed
All of the probabilities are between
0 and 1 and the sum of the
probabilities is 0.9999 ≈ 1.
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) = 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
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
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
Mean, Variance, and Standard Deviation
• Mean: μ = np
• Variance: σ2 = 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: σ2 = 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