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Binomial vs. GeometricChapter

Binomial and Geometric Distributions

Binomial vs. GeometricThe Binomial Setting The Geometric Setting

1. Each observation falls intoone of two categories.

2. The probability of successis the same for each observation.

3. The observations are allindependent.

4. There is a fixed number nof observations.

4. The variable of interest is thenumber of trials required toobtain the 1st success.

1. Each observation falls intoone of two categories.

2. The probability of successis the same for each observation.

3. The observations are allindependent.

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

CombinationsFormula:

Practice:

1.

2.

!" = !!

"! ! − " !

6!4! 6 − 4 ! =

64 = 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1

4 ⋅ 3 ⋅ 2 ⋅ 1(2!) = 6 ⋅ 52 ⋅ 1 = 15

8!5! 8 − 5 ! =

85 = 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1

5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1(3!)= 8 ⋅ 7 ⋅ 63 ⋅ 2 ⋅ 1 = 56

CombinationsFormula:

Practice (continued):3) 5 6) 34) 35 7) 35) 1 8) 1

!" = !!

"! ! − " !

Developing the Formula

Outcomes Probability RewrittenOcOcOc

OOcOcOcOOcOcOcOOOOcOOcOOcOOOOO

O: blood type OOc: complement; not O

! "# = #.&'# ⋅ #. )'"

! "* = #.&'* ⋅ #. )'&

! "& = #.&'& ⋅ #. )'*

! "" = #.&'" ⋅ #. )'#

1⋅

3⋅

3⋅

1⋅

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Developing the Formula

Rewritten

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

! " = $ =%$ &' 1 − & *+'

, -. = ..01. ⋅ .. 31-

, -4 = ..014 ⋅ .. 310

, -0 = ..010 ⋅ .. 314

, -- = ..01- ⋅ .. 31.

1⋅

3⋅

3⋅

1⋅

Working with probability distributions

State the distribution to be usedDefine the variableState important numbers

Binomial: n & pGeometric: p

aka: How to solve these in the AP Statistics exam

1. 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

1a. What is the probability that two used express checkout?

binomialX = # of people use express

n = 5 p = .25

1b. What is the probability that at least four used express checkout?

binomialX = # of people use express

n = 5 p = .25

2.“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

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2a. Interpret P(X = 3) and find the numerical answer.binomialX = # of people who believe

n = 8 p = .60

Interpretation: P(X = 3) is the probability that 3 of the people from the random sample of 8 believe their children will have a higher standard of living.

2b. 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

Developing the Geometric Formula

X Probability

!1 6!5 6 % !1 6

!5 6 % !5 6 % !1 6

!5 6 % !5 6 % !5 6 % !1 6

& ' = ) = * − , )-* % ,

An experiment consists of rolling a single die. The event of interest is rolling a 3; this event is called a success. X is defined as the number of trials until a success occurs.

Formula:

How many dice do we need to roll until we can expect to get a 3?

This is intuitive: 6

Formula:

. ' = *,

1. 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 until defective

p = .03

2. Sophie is a dog who loves to play catch. Unfortunately, she isn’t very good, and the probability that she catches a ball is only .1. We want to count number of tosses required until Sophie catches a ball.

geometricX = # of tosses till catch

p = .10

2a. What is the probability that it will take exactly three tosses for Sophie to catch a ball?

geometricX = # of tosses till catch

p = .10

2b. What is the probability that it will take at most 3 tosses for Sophie to catch a ball?

geometricX = # of tosses till catch

p = .10

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Please complete the rest of the examples as homework.

3. In testing a certain kind of truck tire over a rugged terrain, it is found the 25% of the truck fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability that 4 or 5 will have a blowout.

binomialX = # of tires that fail due to a blowout

n = 15 p = .25

154 + 15

5

Homework Correction: Binomial

4. A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state.

binomialX = # in-state vehicles

n = 9 p = .75

94

a. What is the probability that exactly 4 of the next 9 vehicles are from this state?

binomialX = # out-of-state vehicles

n = 7 p = .25

73

b. What is the probability that exactly 3 of the next 7 vehicles are from out of state?

3. A survey conducted by the Harris polling organization discovered that 80% of all Americans are overweight. Suppose that a number of randomly selected Americans are weighed.

a. What is the probability that the second person weighed is the first person to be overweight?

geometricX = # of persons until overweight

p = .8

Homework Correction: Geometric3b. What is the probability that the fourth person weighed is the first person to be overweight?

Homework Correction: Geometric

3c. What is the probability that that it takes more than four people to observe the first overweight person?

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4. It is estimated that 45% of people in Fast-Food restaurants order a diet drink with their lunch. a. Find the probability that the fourth person orders a diet drink. simple, individual probability

Homework Correction: Geometric

b. Find the probability that the first diet drinker of the day occurs before the 5th person.

geometricX = # of persons until diet drink

p = .45