The Practice of StatisticsThird Edition

Chapter 8:The Binomial and

Geometric Distributions

Copyright © 2008 by W. H. Freeman & Company

Finding Binomial Probabilities

• We will rarely use the formula for finding

the probability of a binomial distribution.

• We will let the graphing calculator do it for

us.

• Look at Example 8.7 on Page 520.

– binompdf

– pdf stands for probability distribution function

binompdf

• Back to the bad switches example

• To find P(X ≤ 1), n = 10, p = .1

• 2nd|(VARS)/0:binompdf (10,.1,0) = .3487

• 2nd|(VARS)/0:binompdf (10,.1,1) = .3874

• So P(X ≤ 1) = .3487 + .3874 = .7361

• 73.61% of all samples will contain no more

than 1 defective switch.

Probability Distribution for

binomial distribution with n = 10

and p = .1 or B(10, .1)

About 74% of all samples will contain no more than 1 bad switch.

A sample size of 10 cannot be trusted to alert the engineer to the

presence of unacceptable items in the shipment.

Doesn’t really matter what the x axis label is… B(10, .1) is always going to

look like this. Every Binomial Distribution has its own look.

http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html

Example 8.8 on page 521.

What is the probability that Corinne will only make at most 7

out of 12 Free Throws, if she is a 75% shooter?

Adding This All up is to Much Work

• P(X ≤ 7) = P(X = 0) + P(X = 1) + P(X + 2)

+…+ P(X = 7) = .1576

• Have to do binompdf for each of these.

• Is there a shorter way?

• of course …

• binomcdf

2nd|(VARS)/A:binomcdf (n, p, X)

n = 12, p = .75, X = 7

2nd|(VARS)/A:binomcdf (12,.75,7) = .1576

So 15.76% of the time Corrine will make no more than

7 out of 12 foul shots. Not unusual.

Remember for calculator:

Binompdf (n,p) Binomcdf (n, p, X)

“Bananapox”

Binomial Mean and Standard

Deviation

• If count X is binomial with n observations and p

probability, what is the mean μ?

– Corinne shoots 12 free throws with 75%

accuracy, we would expect (12)(.75) = 9 to be μ.

• So in general, the mean of a Binomial Random

Variable is μ = np.

• Standard Deviation σ = √(np(1 – p)).

– Trust me on that one.

• Both on formula sheet.

WARNING: THESE FORMULAS ARE

GOOD ONLY FOR BINMOMIAL

DISTRIBUTIONS. THEY CANNOT BE

USED FOR OTHER DISCRETE RANDOM

VARIABLES!!!

Bad Switches

• Sample size n = 10

• Probability of defective switch p = .1

• Mean µ = np = (10)(.1) = 1

• Standard Deviation α = √(np(1 – p))

• = √((10)(.1)(1 - .1)) = .9487

Normal Approximation of

Binomials

• A useful fact – as the number of trials n gets

larger, the binomial distribution gets close

to a Normal distribution. When n is large,

we can use Normal probability calculations

to approximate hard-to-calculate binomial

probabilities

http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html

Attitudes Towards Shopping

• A survey asked a nation-wide random sample of 2500

adults if they agreed or disagreed that “I like buying new

clothes, but shopping is often frustrating and time-

consuming.” Suppose that in fact 60% of all adult U.S.

residents would say “Agree”. What is the probability that

1520 or more of the sample agree?

Independent?

Attitudes Towards Shopping

• A survey asked a nation-wide random sample of 2500

adults if they agreed or disagreed that “I like buying new

clothes, but shopping is often frustrating and time-

consuming.” Suppose that in fact 60% of all adult U.S.

residents would say “Agree”. What is the probability that

1520 or more of the sample agree?

Find the mean and standard deviation.

Attitudes Towards Shopping

• A survey asked a nation-wide random sample of 2500

adults if they agreed or disagreed that “I like buying new

clothes, but shopping is often frustrating and time-

consuming.” Suppose that in fact 60% of all adult U.S.

residents would say “Agree”. What is the probability that

1520 or more of the sample agree?

http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html

Use the normal calculations with this mean and

standard deviation.

Assignment

• “Binomial pdfs and cdfs” worksheet

• “Normal approximation to the binomial distribution” worksheet

• Calculator (you need to know these calculator functions):https://youtu.be/F6JBimUE43U?list=PLkIselvEzpM7N8zVRRUl7V8aTdoT

sJ919

• Read Technology Toolbox on pages 530 – 532 and do on your

calculator.

• Read pages 540 – 548