Binomial vs. Geometric

Chapter 6

Binomial and Geometric

Random Variables

Combinations

Formula:n

k

n

k n k

FHGIKJ

!

! !a fPractice:

16

4. FHGIKJ

6

4 6 4

!

! !a f

6 5 4 3 2 1

4 3 2 1 2 1a f

6 5

2 1 15

28

5. FHGIKJ

8

5 3

!

! !

8 7 6 5

5 3 2 1

!

! 56

Developing the FormulaX

P X( )

Outcomes Probability Rewritten

0

OcOcOc (. )(. )(. )75 75 75

.4219

1

OOcOc

OcOOc

OcOcO

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

.4219

2

OOOc

OOcOOcOO

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

.1407

3

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

.0156

1 .253a f

3 . .25 752a f a f1

3 . .25 751 2a f a f

.750a f

1 25 750 3

. .a f a f3

0

FHGIKJ

3

1

FHGIKJ

3

2

FHGIKJ

3

3

FHGIKJ

Developing the Formula

Rewritten

1 .253a f

3 . .25 752a f a f1

3 . .25 751 2a f a f

.750a f

1 25 750 3

. .a f a f3

0

FHGIKJ

3

1

FHGIKJ

3

2

FHGIKJ

3

3

FHGIKJ

n = # of observations

p = probablity of success

k = given value of variable

P X k( ) n

k

FHGIKJ pk

1

pn ka f

Binomial vs. GeometricThe Binomial Setting The Geometric Setting

What is

Binomial???

Just check your

BINS

What is

Geometric???

Just check your

BITS

Binomial vs. GeometricThe Binomial Setting The Geometric Setting

1. Binary: Only two possible

Outcomes (Success/Fail)

2. Independent: The

observations are all

independent

3. Number: There is a fixed

number of trials

4. Success: The probability

of success is the same for

each trial

4. Success: The probability

of success is the same for

each trial

1. Binary: Only two possible

outcomes (Success/Fail)

2. Independent: The

observations are all

independent

3. Trials: We go until the first

success

Working with probability distributions

State the distribution to be used

Binomial or Geometric

Define the variable

X = number of …

State important numbers

Binomial: n & p

Geometric: p

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.

binomial

X = # of people use express

n = 5 p = .25

What is the probability that two used express checkout?

binomial

X = # 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?

binomial

X = # of people use express

n = 5 p = .25

4P X 5

4

4

.25 1

.75

.0156

55

.255

“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.

binomial

X = # of people who believe…

n = 8 p = .60

Interpret P(X = 3) and find the numerical answer.

binomial

X = # 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 will

have a higher standard of living.

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

binomial

X = # of people who believe

n = 8 p = .60

0P X 8

0

0

.6 8

.4 .0007

The Mean and Standard Deviation of a Binomial Random

VariableIf X is a binomial random variable with probability of success p on each trial and n number of trials, the expected value and std. dev. of the random variable is

x np (1 )x np p

Developing the Geometric Formula

X Probability

1 16

25

61

6d id i3 5

65

61

6d id id i4

56

56

56

16d id id id i

P X n a f 11

pna f p

The Mean of a Geometric Random Variable

If X is a geometric random variable with probability of success p on each trial, the expected value of the random variable (the expected number of trials to get the first success) is

1

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.

geometric

X = # of disc drives until 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.

geometric

X = # of free throws until miss

p = .20

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

geometric

X = # of free throws until miss

p = .20

6P X 5

.80 .20 .0655

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

geometric

Y = # of free throws until make

p = .80

6P Y 5

.20 .80 .00026

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

geometric

X = # of free throws until miss

p = .20

6P X .20 .80 .20

.7379

2

.80 .20

3

.80 .20 4

.80 .20 5

.80 .20