AP Statistics: Section 8.1A

Binomial Probability

There are four conditions to a binomial setting:

1. Each observation falls into one of just two categories: _________ or ________.

2. There is a ______ number of observations, __.

3. These n observations are all ____________.

4. The probability of success, __, is _________ for each observation.

failure success

finite n

tindependen

constant p

If data are produced in a binomial setting, then the random variable X = the number of successes is called a binomial random

variable and the probability distribution of X is called a binomial distribution which is

abbreviated ______.),( pnB

Example 1: Determine if each of the following situations is a

binomial setting. If so, state the probability distribution for X. If

not, state which of the 4 conditions above is not met.

Situation 1: Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so having blood type O. A couple’s 5 different children inherit

independently of each other. Let X = number of children with type O blood.

)25,.5(B

Situation 2: Deal 10 cards from a shuffled deck and let X = the number of red cards.

51

25

51

26,

52

26

constant.not are iesprobabilit No,

or

Situation 3: An engineer chooses a SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. Let X = the number of bad switches in the sample.

)1,.10(B

****When choosing an SRS from a population that is much larger than the sample, the observations are considered independent.

Binomial Probability: If X has a binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, 3, . . . , n. If k is any one of these values, then

knk ppk

nkxP

)1)(()(

The notation is read n choose k and means the number of possible

ways to choose k objects from a group of n objects.

It is also written _____

k

n

knC

:84/83TI

k

n

)!(!

!

knk

n

rCn: 3 PRB MATH

(3)(2)(1)2)-1)(n-n(n means and factorialn read is n!notation The

720123456 6! So

10! definitionBy

! :4 PRB MATH :84/83TI

Example 2 (combinations): How many different ways can we choose a

subcommittee of size 3 from a student council that has 7 members?

356

210

!4!3

!7

3

7

3537 rnC

Example 3: You randomly guess the answers to10 multiple choice questions which have 5 possible answers. What is the probability of

getting exactly 6 correct answers?

6 with x)2,.10( B

)8)(.2(.

6

10 46 0055.

Binomial Probability on the TI83/84:

ENTER fbinomialpd:0 DISTR VARS 2nd

),,( xpnfbinomialpd

Example 4: Consider situation 3 in example 1. Find the probability that in an SRS of size 10, no

more than 1 switch fails.

x where)1,.10( B 1or 0

.7361

3874.)1,1,.10(

3487.)0,1,.10(

fbinomialpd

fbinomialpd

Example 5: Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a big game, Corinne shoots 12 free throws and makes only 5 of them. Is it unusual for Corinne to perform this poorly?Note: We actually want the probability of making a basket on at most 5 free throws.

Cumulative Binomial Probability on the TI83/84:

5or 0,1,2,3,4 x where)75,.12( B

fbinomialcd:A DISTR VARS 2nd

value)largest x ,,( pnfbinomialcd

0143.)5,75,.12( binmialcdf

Any difference in answers between the manual calculation and using your calculator is due to rounding

error.