Binomial vs. Geometric

Chapter 8Binomial 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.

For each of the following situations, determine if the situation is binomial,

geometric, or neither. 1. You observe the gender of the next 20 children born

at a local hospital; X is the number of girls among them.

2. Bobby draws cards from a deck until he gets an ace. He then reshuffles, and starts over and repeats this process until he “wins” 10 times. The count X is the total number of cards he counted until he finished.

3. Joe buys a “Texas 2-Step” lottery ticket every week until he wins. He has a probability of .0023 of winning in any given week.

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.

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

Developing the Binomial FormulaX

P X( )

Outcomes Probability Rewritten

0

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

.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

P(O) = .25

P(Oc) = .75

Developing the Binomial 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 = probablity of successk = given value of variable

P X k( ) n

kFHGIKJ p

k 1 p n ka f

The Mean and Standard Deviation of a Binomial Random

Variable

If a count X has the binomial distribution with number of observations n and probability of a success p…

Mean:

Standard Deviation:

np

1np p

Probability Distribution Function (pdf)

Given a discrete random variable X, the pdf assigns a probability to each value of X.

Cumulative distribution Function (cdf)

Given a random variable X, cdf calculates the sum of the probabilities for 0, 1, 2, …, X.

It calculates the probability of obtaining at most X successes in “n” trials.

Calculator Functions:pdf: means X = ?

2nd Distr. -> 0 (binompdf)

binompdf(n, p, X)

cdf: means X ≤ ?

2nd Distr. -> A (binomcdf)

binomcdf(n, p, X)

Ex 1. The count X of children with type O blood among 5 children whose parents carry genes for both O and A blood types is B(5, .25).

Find P(X = 2).

Find P(X < 4)

Ex 2. Angel is a basketball player who makes 75% of her free throws over the course of a season. At last night’s game, she

missed 5 of 7 free throws. Assume that each free throw is independent of one another…studies on long sequences of

free throws have found no evidence that they are dependent.

Define the variable.

Complete the table.-----------------------------------------------------------------------X-----------------------------------------------------------------------P(X)-----------------------------------------------------------------------F(X)

What is the probability that Angel makes at most 6 free throws in a

row?

What is the probability that Angel makes more than 3 free throw shots?

Find the mean and Standard Deviation of the following random

variable.

According to a recent Census Bureau report, 12.7% of Americans live below the poverty level. Suppose you plan to sample at random 100 Americans and count the number of people who live below the poverty level.

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

16

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

Probability of a Success: P(O) =

Probability of a Failure: P(Oc) = 65

X = wanted outcome O

The Mean and Standard Deviation of a Geometric

Random VariableIf 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

1p

1

2p

p

Calculator Functions:pdf: means X = ?

2nd Vars (Distr.) -> D (geometpdf)geometpdf(p, X)

cdf: means X ≤ ?2nd Vars (Distr.) -> E (geometcdf)geometcdf(p, X)

Ex 2. Creating a Geometric Probability Distribution table for X = number rolls of a die

until a 4 occurs:

The probability of rolling a 4 on die =

A geometric probability does not have a fixed ending value “n”, therefore it is considered an infinite sequence. Meaning the “x” goes on infinitely.

X 1 2 3 4 5 …

P(X) …1

6

1

6

5

36

25

216

Ex 2. What is the probability that it takes more than 6 rolls to observe a

4?

Ex 3. What is the probability that it takes at least 3 rolls to observe a 4?

Find the mean and Standard Deviation of the following random

variable. A potential buyer will sample videotapes from a large lot of new videotapes. If she finds at least one defective one, she’ll reject the entire lot. If ten percent of the lot is defective, what is the probability that she’ll find a defective tape by the 4th videotape?