Bluman, Chapter 5 1

guessing

Suppose there is multiple choice quiz on

a subject you don’t know anything

about…. 15th Century Russian Literature;

Nuclear physics etc.

You have to guess on every question.

There are 5 questions and each question

has 4 choices.

Bluman, Chapter 5 2

Let x be the score on the test.

Find p(x=0)

In another words the probability you will get

a score of zero, i.e. you will get all the

questions wrong

Find p(x=1)

In another words the probability you will get

a score of 1, i.e. you will get only one

question correct.

Bluman, Chapter 5 3

Bluman, Chapter 5 4

Question

number

Correct or wrong

1

2

3

4

5

Repeat the process:

P(2)=

P(3)=

p(4)=

P(5)=

Bluman, Chapter 5 5

What if the number of questions

changed

Let’s say now the test has 10 questions

and each question has 4 choices.

What does the probability distribution chart

looks like?

Bluman, Chapter 5 6

Bluman, Chapter 5 7

x P(x)

0

1

2

3

4

5

6

7

8

9

10

What if the number of choices

changes

Let’s say now the test has 10 questions

and each question has 5 choices.

What does the probability distribution chart

looks like?

Bluman, Chapter 5 8

Bluman, Chapter 5 9

1

2

3

4

5

6

7

8

9

10

5-3 The Binomial Distribution

10

Many types of probability problems have

only two possible outcomes or they can be

reduced to two outcomes.

Examples include:

when a coin is tossed it can land on heads or

tails,

when a baby is born it is either a boy or girl.

It will rain or it won’t

A person will pass the bar exam or not.

The Binomial Distribution

Bluman, Chapter 5 11

The binomial experiment is a probability

experiment that satisfies these requirements:

1. Each trial can have only two possible

outcomes—success or failure.

2. There must be a fixed number of trials.

3. The outcomes of each trial must be

independent of each other.

4. The probability of success must remain the

same for each trial.

Notation for the Binomial Distribution

Bluman, Chapter 5 12

The symbol for the probability of success

The symbol for the probability of failure

The numerical probability of success

The numerical probability of failure

and P(F) = 1 – p = q

The number of trials

The number of successes

P(S)

P(F)

p

q

P(S) = p

n

X

Note that X = 0, 1, 2, 3,...,n

The Binomial Distribution

!

- ! !

X n XnP X p q

n X X

Bluman, Chapter 5 13

In a binomial experiment, the probability of

exactly X successes in n trials is

number of possible probability of adesired outcomes desired outcome

or

X n X

n xP X C p q

Chapter 5

Discrete Probability Distributions

Section 5-3

Example 5-16

Page #272

Bluman, Chapter 5 14

Example 5-16: Survey on Doctor Visits

A survey found that one out of five Americans say

he or she has visited a doctor in any given month.

If 10 people are selected at random, find the

probability that exactly 3 will have visited a doctor

last month.

Bluman, Chapter 5 15

!

- ! !

X n XnP X p q

n X X

3 7

10! 1 43

7!3! 5 5

P

15

10,"one out of five" , 3 n p X

0.201

Chapter 5

Discrete Probability Distributions

Section 5-3

Example 5-17

Page #273

Bluman, Chapter 5 16

Example 5-17: Survey on Employment A survey from Teenage Research Unlimited

(Northbrook, Illinois) found that 30% of teenage

consumers receive their spending money from

part-time jobs. If 5 teenagers are selected at

random, find the probability that at least 3 of them

will have part-time jobs.

Bluman, Chapter 5 17

3 25!

3 0.30 0.702!3!

P

5, 0.30,"at least 3" 3,4,5 n p X

0.132

4 15!

4 0.30 0.701!4!

P 0.028

5 05!

5 0.30 0.700!5!

P 0.002

3 0.132

0.028

0.002

0.162

P X

Chapter 5

Discrete Probability Distributions

Section 5-3

Example 5-18

Page #273

Bluman, Chapter 5 18

Example 5-18: Tossing Coins

A coin is tossed 3 times. Find the probability of

getting exactly two heads, using Table B.

Bluman, Chapter 5 19

12

3, 0.5, 2 n p X 2 0.375 P

The Binomial Distribution

Mean: np

2Variance: npq

Bluman, Chapter 5 20

The mean, variance, and standard deviation

of a variable that has the binomial distribution

can be found by using the following formulas.

Standard Deviation: npq

Chapter 5

Discrete Probability Distributions

Section 5-3

Example 5-23

Page #276

Bluman, Chapter 5 21

Example 5-23: Likelihood of Twins The Statistical Bulletin published by Metropolitan

Life Insurance Co. reported that 2% of all American

births result in twins. If a random sample of 8000

births is taken, find the mean, variance, and

standard deviation of the number of births that

would result in twins.

Bluman, Chapter 5 22

8000 0.02 160 np

2 8000 0.02 0.98 156.8 157 npq

8000 0.02 0.98 12.5 13 npq

Tech notes

Read technology

notes on page

281.

Read example 5-

19 on page 274

Exercises 5.3

Page 276 #1,

5, 11, 15 and

17

Bluman, Chapter 5 23