Section 4.2 Binomial Distributions Larson/Farber 4th ed 26

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


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.


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.


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.

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.

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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)


Binomial Probability Formula

35Larson/Farber 4th ed

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

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


Solution: Finding Binomial Probabilities

Method 1: Draw a tree diagram and use the Multiplication Rule

37


Method 2: Binomial Probability Formula


= 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

Use binomial probability formula to find probabilities.


x 0 1 2 3 ...

P(x) 0.283 0.377 0.230 0.085 ...

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

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

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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

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.


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).



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

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

Example: Finding Binomial Probabilities Using a Table

# 26 on page 218 of the book

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Solution:• Binomial: n = 5, p = 0.25, q = 0.75, x = 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

x Probability

0

1

2

3

4

5

Mean, Variance, and Standard Deviation

• Mean: μ = np

• Variance: σ2 = npq

• Standard Deviation:


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.

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Solution: n = 12, p = 0.14, q = 0.86

Mean: μ = np = (12)∙(0.14) = 1.68Variance: σ2 = npq = (12)∙(0.14)∙(0.86) ≈ 1.45Standard 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


Documents

Section 4.2 Binomial Distributions Larson/Farber 4th ed 26