Another Random Variable

Suppose that each of three randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. The number among the three customers who purchase an electric hot tub is a random variable. What is the probability distribution?

Binomial vs. Geometric

Chapter 17Binomial and Geometric

Distributions

Binomial vs. GeometricThe Binomial Setting The Geometric Setting

1. Each observation falls into one of two categories.

2. The probability of success is the same for each observation.3. The observations are all independent.

4. There is a fixed number n of observations.

4. The variable of interest is the number of trials required to obtain the 1st success.

1. Each observation falls into one of two categories.2. The probability of success is the same for each observation.

3. The observations are all independent.

Are Random Variables and Binomial Distributions Linked?

XP(X)

X = number of people who purchase electric hot tub

0 1 2 3

GGG (.6)(.6)(.6)

.216

EGGGEGGGE

(.4)(.6)(.6)(.6)(.4)(.6)(.6)(.6)(.4)

.432

EEGGEEEGE

(.4)(.4)(.6)(.6)(.4)(.4)(.4)(.6)(.4)

.288

EEE (.4)(.4)(.4)

.064

Combinations

Formula:n

kn

k n kFHGIKJ

!

! !a fPractice:

16

4. FHGIKJ

64 6 4

!! !a f

6 5 4 3 2 14 3 2 1 2 1a f

6 52 1

15

28

5. FHGIKJ

85 3

!! !

8 7 6 55 3 2 1

!!

56

M&MsAccording to Mars a few years ago, 25% of M&Ms are orange.You pull three M&Ms out of a bag.

Developing the FormulaX

P X( )

Outcomes Probability Rewritten

0

OcOcOc

.42191

OOcOc

OcOOc

OcOcO

(. )(. )(. )25 75 75

.42192

OOOc

OOcOOcOO

(. )(. )(. )25 25 75

.14063

OOO (. )(. )(. )25 25 25

.0156

1 .25 3af3 . .25 752afaf1

3 . .25 751 2afaf

.75 0af

1 25 750 3. .afaf3

0FHGIKJ

3

1FHGIKJ

3

2FHGIKJ

3

3FHGIKJ

Developing the Formula

Rewritten

1 .25 3af3 . .25 752afaf1

3 . .25 751 2afaf

.75 0af

1 25 750 3. .afaf3

0FHGIKJ

3

1FHGIKJ

3

2FHGIKJ

3

3FHGIKJ

n = # of observationsp = probability of successk = given value of variable

P X k( ) n

kFHGIKJ p

k 1 p n ka f

Working with probability distributions

State the distribution to be usedDefine the variableState important numbers

Binomial: n & pGeometric: pNormal: &

Twenty-five percent of the customers entering a grocery store between 5 p.m. and 7 p.m. use an express checkout. Consider five randomly selected customers, and let X denote the number among the five who use the express checkout.

binomialX = # of people use express

n = 5 p = .25

What is the probability that two used express checkout?

binomialX = # of people use express

n = 5 p = .25

2P X 5

2

2.25 3

.75 .2637

What is the probability that at least four used express checkout?

binomialX = # of people use express

n = 5 p = .25

4P X 5

4

4.25 1

.75

.0156

05 75.25.5

5

“Do you believe your children will have a higher standard of living than you have?” This question was asked to a national sample of American adults with children in a Time/CNN poll (1/29,96). Assume that the true percentage of all American adults who believe their children will have a higher standard of living is .60. Let X represent the number who believe their children will have a higher standard of living

from a random sample of 8 American adults. binomialX = # of people who believe…

n = 8 p = .60

Interpret P(X = 3) and find the numerical

answer.

binomialX = # of people who believe

n = 8 p = .60

3P X 8

3

3.6 5

.4 .1239

The probability that 3 of the people from the random sample of 8 believe their children willhave a higher standard of living.

Find the probability that none of the parents believe their children will have a higher standard.

binomialX = # of people who believe

n = 8 p = .60

0P X 8

0

0.6 8

.4 .00066

Binomial vs. GeometricThe Binomial Setting The Geometric Setting

1. Each observation falls into one of two categories.

2. The probability of success is the same for each observation.3. The observations are all independent.

4. There is a fixed number n of observations.

4. The variable of interest is the number of trials required to obtain the 1st success.

1. Each observation falls into one of two categories.2. The probability of success is the same for each observation.

3. The observations are all independent.

Developing the Geometric Formula

X Probability

1 16

2 56

16didi

3 56

56

16didid i

4 56

56

56

16didid id i

P X n a f 1 1 p na f p

Suppose we have data that suggest that 3% of a company’s hard disc drives are defective. You have been asked to determine the probability that the first defective hard drive is the fifth unit tested.

geometricX = # of disc drives till defective

p = .03

5P X 4.97 .03 .0266

A basketball player makes 80% of her free throws. We put her on the free throw line and ask her to shoot free throws until she misses one. Let X = the number of free throws the player takes until she misses.

geometricX = # of free throws till miss

p = .20

What is the probability that she will make 5 shots before she misses?

geometricX = # of free throws till miss

p = .20

6P X 5.80 .20 .0655

What is the probability that she will miss 5 shots before she makes one?

geometricY = # of free throws till make

p = .80

6P Y 5.20 .80 .00026

What is the probability that she will make at most 5 shots before she misses?

geometricX = # of free throws till miss

p = .20

6P X .20 .80 .20

.7379

2.80 .20

3.80 .20 4

.80 .20 5.80 .20

What is the probability that she will make at least 8 shots before she misses?

geometricX = # of free throws till miss

p = .20

Binomial formulas for Binomial formulas for mean and standard mean and standard

deviationdeviation

pnp

np

x

x

1

In a certain county, 30% of the voters are Republicans. How many Republicans would you expect in ten randomly selected voters?

What is the standard deviation for this distribution?

sRepublican45.1)7)(.3(.10

sRepublican3)3(.10

x

x

Geometric formulas for Geometric formulas for mean and standard mean and standard deviationdeviation

2

1

1

p

p

p

x

x

A real estate agent shows a house to prospective buyers. The probability that the house will be sold to the person is 35%. What is the probability that the agent will sell the house to the third person she shows it to?How many prospective buyers does she expect to show the house to before someone buys the house?

1479.35.65.)3( 2 xP

buyers86.235.1

x