Larson/Farber Ch. 4

Binomial Distributions

Section 4.2

Larson/Farber Ch. 4

Also:• p + q = 1

• The random variable x is a count of the number of successes in n trials.

• The central problem is to find the probability of x successes out of n trials. Where x = 0 or 1 or 2 … n.

Binomial Experiments4 Conditions for a Binomial Experiment1. There are a fixed number of trials. (n)2. Each trial is independent.3. Each trial has 2 outcomes (p = Success or q =

Failure.)4. The probability of success for each trial is the same.

Larson/Farber Ch. 4

Is it a binomial experiment?

1. An experiment in which a basketball player who historically makes 80% of his free throws is asked to shoot 3 free throws and the number of made free throws is recorded.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?2. The number of people with blood type O-negative based upon a

simple random sample of size 10 is recorded. According to the Information Please Almanac, 6% of the human population is blood type 0-negative.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?3. A probability experiment in which three cards are drawn from a deck

without replacement and the number of aces is recorded.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?4. A random sample of 15 college seniors is conducted, and the

individuals selected are asked to state their ages.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?5. An experimental drug is administered to 100 randomly selected

individuals, with the number of individuals responding favorably recorded.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?6. A poll of 1200 registered voters is conducted in which the

respondents are asked whether they believe Congress shoulc reform Social Security.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?7. A baseball player who reaches base safely 30% of the time is

allowed to bat until he reaches base safely for the third time. The number of at-bats required is recorded.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

Is it a binomial experiment?8. An investor randomly purchases 10 stocks listed on the New York

Stock Exchange. Historically, the probability that a stock listed on the NYSE will increase in value over the course of a year is 48%. The number of stocks that increase in value is recorded.

There are a fixed number of trials?Each trial is independent?Each trial has 2 outcomes? The probability of success for each trial is the same?

If yes:

p = _____ q = _____

n = _____ x = _______________

Larson/Farber Ch. 4

1. If you have three children, what is the probability that you will have none, one, two, or three boys?

How do you construct Binomial Probability Distributions?

Old Method:

1. Construct a tree diagram.

2. Compute probabilities.

New Method:

1. Find x, n, p, and q.

2. Complete binomial distribution table using formula

n xx n xC p q( )

Larson/Farber Ch. 4

Constructing Binomial Distributions

XNumber of boys

P(x) =

0

1

2

3

1. If you have three children, what is the probability that you will have none, one, two, or three boys?

X = _____ p = _____ q = _____ n = _____

n xx n xC p q

Larson/Farber Ch. 4

Constructing Binomial Distributions

X P(x) =

Example 2 - #9 You are taking a multiple-choice quiz that consists of five questions. Each question has four possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each question.

X = _____ p = _____ q = _____ n = _____

n xx n xC p q

Larson/Farber Ch. 4

Constructing Binomial DistributionsExample 2 - #9 cont. Find the probability of the following:a. Exactly 3 answers correctlyb. At least 3 answers correctlyc. Less than 3 answers correctly

Xx X P(x)

Larson/Farber Ch. 4

1. If you have three children, what is the probability that you will have none, one, two, or three boys?

How do you find the Mean and Standard Deviation?

X = 0,1,2,3 p = .5 q = .5 n = 3

x P(x)nCxpxq(n-x)

xP(x) (x-)2P(x)

0

1

2

3

Larson/Farber Ch. 4

• Page 184, #17

x P(x) xP(x) (x-μ)2P(x)

Larson/Farber Ch. 4

Homework

Page 184

#5-8, 16, 18, 20