Section 8.4 The Binomial Distribution

Binomial Experiment

A binomial experiment has the following properties:

1. The number of trials in the experiment is fixed.

2. There are two outcomes of each trial: “success” and “failure.”

3. The probability of success in each trial is the same.

4. The trials are independent of each other.

1. Which of the following experiments are binomial?

(a) Rolling a fair die 20 times and, each time, observing if a 2 appears.

(b) Selecting 4 cards from a standard deck of 52 cards without replacement and, each time,

observing if an ace is drawn.

(c) Assuming that there are 5 yellow and 7 white marbles in a jar. Choosing three marbles from

the jar with replacement (put the marble back in the jar) and, each time, observing if a

yellow marble is choosen.

Computing Probabilities in a Binomial Experiment: Exactly x successes

In a binomial experiment in which the probability of success in any trial is p, the probability of

exactly x successes in n independent trials is given by

P (X = x) = C(n, x)px(1� p)n�x

Computing Probabilities in a Binomial Experiment: At Most x successes

In a binomial experiment in which the probability of success in any trial is p, the probability of

at most x successes in n independent trials is given by

P (X x) = P (X = 0) + P (X = 1) + . . . P (X = x)

Calculator Steps:

2ND , VARS , scroll down to A or click ALPHA , MATH . Your screen should show

binompdf( , this is for exactly x successes. For at most x successes, scoll down to B or

click ALPHA , APPS . This time your screen should show binomcdf( .

Here’s the format for both: binompdf(n, p, x) or binomcdf(n, p, x) .

2. It is estimated that one third of the general population has blood type A+. A sample of six

people is selected at random. (Round answers to four decimal places.)

(a) What is the probability that exactly two of them have blood type A+?

(b) What is the probability that at most two of them have blood type A+?

3. The probability that a DVD player produced by VCA Television is defective is estimated to be

0.06. A sample of ten players is selected at random. (Round answers to four decimal places.)

(a) What is the probability that the sample contains no defective units?

(b) What is the probability that the sample contains at most two defective units?

2 Fall 2017, Maya Johnson

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A Few Di↵erent Ways to use binomcdf

1. P (X � x) = 1� binomcdf(n, p, x� 1) “at least x successes.”

2. P (x1 X x2) = binomcdf(n, p, x2)� binomcdf(n, p, x1 � 1) “at least x1 but at most x2

successes.”

3. P (x1 < X x2) = binomcdf(n, p, x2)� binomcdf(n, p, x1) “more than x1 but at most x2

successes.”

4. P (x1 X < x2) = binomcdf(n, p, x2�1)� binomcdf(n, p, x1�1) “at least x1 but fewer than

x2 successes.”

5. P (x1 < X < x2) = binomcdf(n, p, x2� 1)� binomcdf(n, p, x1) “more than x1 but fewer than

x2 successes.”

4. From experience, the manager of Kramer’s Book Mart knows that 50% of the people who are

browsing in the store will make a purchase. What is the probability that among ten people who

are browsing in the store, at least four will make a purchase? (Round answer to four decimal

places.)

5. Suppose 30% of the restaurants in a certain part of a town are in violation of the health code. A

health inspector randomly selects five of the restaurants for inspection. (Round answers to four

decimal places.)

(a) What is the probability that none of the restaurants are in violation of the health code?

(b) What is the probability that one of the restaurants is in violation of the health code?

(c) What is the probability that at least two of the restaurants are in violation of the health

code?

3 Fall 2017, Maya Johnson

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6. The manager of Toy World knows that the probability an electronic game will be returned to the

store is 0.22. If 54 games are sold in a given week, determine the probabilities of the following

events. (Round answers to four decimal places.)

(a) No more than 12 games will be returned.

(b) At least 8 games will be returned.

(c) More than 5 games but fewer than 14 games will be returned.

7. A coin is biased so that the probability of tossing a head is 0.46. If this coin is tossed 54 times,

determine the probabilities of the following events. (Round answers to four decimal places.)

(a) The coin lands heads more than 21 times.

(b) The coin lands heads fewer than 28 times.

(c) The coin lands heads at least 20 times but at most 27 times.

4 Fall 2017, Maya Johnson

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p = a46

, na 54

P( X > 21 ) = 1 - Binomcdf ( 54 ,a 46

,21)=@9 ,

" don't want 21,

so subtract it off "

P ( X < 28 ) = Binomcdf ( 54 , .46 ,27

) = .7@" don't warn

't 28 ,so go

down by1

"

P ( 204427 ) = Binomcdf ( 54 , . 46,27 ) -Binomcdf 154 , . 46 ,

19 )

= @" Wan 't 20 , so go down by 1

"

8. A company finds that one out of four employees will be late to work on a given day. If this

company has 40 employees, find the probabilities that the following number of people will get to

work on time. (Round answers to four decimal places.)

(a) Exactly 31 workers or exactly 35 workers.

(b) At least 27 workers but fewer than 35 workers.

(c) More than 25 workers but at most 37 workers.

Mean, Variance, and Standard Deviation for Binomial Distribution

If X is a binomial random variable associated with a binomial experiment consisting of n trials

with probability of success p, then the mean, variance, and standard deviation of X are

µ = E(X) = np

V ar(X) = np(1� p)

�

x

=p

np(1� p)

9. A new drug has been found to be e↵ective in treating 70% of the people a✏icted by a certain

disease. If the drug is administered to 500 people who have this disease, what are the mean,

variance, and the standard deviation of the number of people for whom the drug can be expected

to be e↵ective? (Round answers to two decimal places.)

5 Fall 2017, Maya Johnson

f- 314 ,n = 40

Pl Xa 31 ) + P ( X = 35 ) = Binompdf ( 40,314,31 )t Binompdf140,314,357

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P( 27 4- X L 35 ) = Binomcdf ( 40,314,34 ) - Binomcdf (40,3/4,26)" wan 't 27

, godown by

1"

=€" don't won't

:40,314 ,

37 ) - Binomcdf (40,3/4,25)P ( 25 L X E 37 ) = Binomcdt a don't wan

't 25"

2.94€

of success .

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= 10.250