Chapter 8:The Binomial and

Geometric Distributions

Copyright © 2008 by W. H. Freeman & Company

The Practice of StatisticsThird Edition

Random Phenomena with Two

Possible Outcomes

• Coin toss

– Heads/Tails

• Shooting a free throw.

– Make/Miss

• The birth of a child.

– Boy/Girl

• These are called Binomial Distributions

A discrete random variable situation can be

a binomial setting, a geometric setting (to

be covered later in the chapter), or neither.

You need to be able to recognize which

situation it is.

If data are produced in a binomial setting, then the

random variable X = number of success. This is called a

BINOMIAL RANDOM VARIABLE, and the

probability distribution of X is called a BINOMIAL

DISTRIBUTION.

Caution• Binomial distributions are an important

class of discrete probability distributions.

• The most important skill for

using binomial distributions is

the ability to recognize

situations to which they do and

don’t apply!!!

Examples

• Blood types

– If both parents carry genes for O and A type

blood, each child has probability .25 of

inheriting type O blood.

– If couple has 5 children, the count X is the

number of success of the 5 children that have

type O blood.

– n = 5 and p = .25

– B(5, .25)

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

Example

• Dealing Cards

– Deal 10 cards and count the number of red

cards.

– Not independent

– If first card is red, it is more likely the second

card is black.

– Not a binomial distribution

Binomial Distributions

in Statistical Sampling

• Important because we use this to make

inferences the proportion “p” of success in a

population.

• Choose an SRS of n = 10 switches from 10,000

switches. 10% of switches are bad so p = 0.1.

– Not a binomial setting.

– However, with large samples, not having

independence is not that big of a deal.

Assume that airplane engines have probability 0.999 of performing

properly for an hour of flight.

350 engines that fly for an hour without failure has a

B(350, 0.999) distribution.

This assumes engines fail independently.

Binomial Formulas

• We can find a formula for the probability

that a binomial random variable takes any

value by adding probabilities for the

different ways of getting exactly that many

successes in n observations.

Inherit Blood Type

• Child has .25 probability of having type O

blood.

• Parents have 5 children

• What is probability of exactly 2 of them

having type O blood?

• n = 5 and p = .25

• We want P(X = 2)

Binomial Formulas

• S = Success (what does this mean in this context?)

• F = Failure

• Find the probability of a specific 2 out of 5.

• As an example:

– SFSFF (1st and 3rd have type O blood)

• (.25)(.75)(.25)(.75)(.75) = (.25)2(.75)3

• How many possible combinations of 2 success and 3

failures are there?

Blood Types

• There are 10 possible combinations of 2

Successes and 3 Failures.

– How do I know this?

– We will find out on the next slide!

• They all have the same probability.

• So: P(X=2) = 10(.25)2(.75)3 = .2637

n = 5, k = 2

n! = 5*4*3*2*1 = 120

k! = 2*1 = 2

(n – k)! = (5 – 2)! = 3! = 3*2*1 = 6

120/(2*6) = 120/12 = 10 Combinations

You should learn

this – not on

formula sheet!

Read aloud "n choose k."

A binomial coefficient equals

the number of combinations

of k items that can be

selected from a set of n items.

OR:

On your calculator, enter n and then Math:PRB:nCr and then

enter k and then ENTER.

Binomial Probability

Defective Switches

n = 10, and p = .1

Probability that no more than 1 switch fails.

P(X ≤ 1) = P(X = 1) + P(X = 0)

In this case n = 10 and k = 1

Use what you just learned to try to do this.

P(X = 1) = .3874 and P(X = 0) = .3487

P(X ≤ 1) = .3874 + .3487 = .7361

Assignment

• Play with Binomial Distributions Applet at

http://homepage.divms.uiowa.edu/~mbognar/applets/bin.html and see when it

becomes approximately normal.

• “How to Calculate Binomial Probabilities” Packet

• Calculator videos that you need to know:

Binomial coefficient (Combinations):

https://youtu.be/MgSitJ7Aqxg?list=PLkIselvEzpM7N8zVRRUl7V8aTdoTsJ919

Binomial formula (just watch until 1:19 for now): https://youtu.be/F6JBimUE43U?list=PLkIselvEzpM7N8zVRRUl7V8aTdoTsJ919

• Need a review?

Watch https://youtu.be/mKtNpY1ZEdw?list=PLC8478000586FA6F9

• Read Pages 517 – 526