Binomial Distributions

Binomial Experiments• Probability experiments for which the

results of each trial can be reduced to two outcomes: success and failure.

• When a basketball player attempts a free throw, he or she either makes the basket or does not.

• Probability experiments such as these are called binomial experiments.

Binomial Probability Distribution A Fixed Number of Observations (trials), n

15 tosses of a coin; 20 patients; 1000 people surveyed

A Binary Outcome Head or tail in each toss of a coin; disease or no

disease Generally called “success” and “failure” Probability of success is p, probability of failure is

1 – p

Constant Probability for each observation Probability of getting a tail is the same each time we

toss the coin

Notation for Binomial Experiments

Symbol Description

n The number of times a trial is repeated.

p = P(S) The probability of success in a single trial.

q = P(F) The probability of failure in a single trial (q = 1 – p)

X The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, . . . n.

Example

• We pick a card from a standard deck of cards, and note whether it is a club or not, and replace the card. We repeat the experiment 5 times.

• n = 5• p = P(S) = ¼• q = P(F) = ¾• Possible values of the random variable are

0, 1, 2, 3, 4, and 5.

Binomial Experiments• Decide whether the experiment is a

binomial experiment:

• A binomial experiment specify the values

of n (number of times a trial is repeated), p

(Probability of Success), q (Probability of

Failure) and list the possible values of the

random variable, x.

Example

A certain surgical procedure has an

85% chance of success. A doctor

performs the procedure on eight

patients. The random variable

represents the number of successful

surgeries.

Binomial Experiment• Each surgery represents one trial. There

are eight surgeries, and each surgery is independent of the others.

• Only two possible outcomes for each

surgery - either the surgery is a success or it is a failure.

n = 8p = 0.85q = 1 – 0.85 = 0.15x = 0, 1, 2, 3, 4, 5, 6, 7, 8

Example

A jar contains five red marbles, nine blue

marbles and six green marbles. Select

randomly three marbles from the jar, without

replacement. The random variable

represents the number of red marbles.

Not A Binomial Experiment• Each marble selection represents one trial

and selecting a red marble is a success. • When selecting the first marble, the

probability of success is 5/20. However because the marble is not replaced, the probability of further trials is no longer 5/20.

• Trials are not independent.

Binomial Probability Formula

xnxxnxx

n qpxxn

nqpCxP

!)!(

!)(

Example: Binomial Probabilities

• A six sided die is rolled 3 times. Find the

probability of rolling exactly one 6.

Roll 1 Roll 2 Roll 3 Frequency # of 6’s Probability

(1)(1)(1) = 1 3 1/216

(1)(1)(5) = 5 2 5/216

(1)(5)(1) = 5 2 5/216

(1)(5)(5) = 25 1 25/216

(5)(1)(1) = 5 2 5/216

(5)(1)(5) = 25 1 25/216

(5)(5)(1) = 25 1 25/216

(5)(5)(5) = 125 0 125/216

Example: Binomial Probabilities• Three outcomes that have exactly one six• Each has a probability of 25/216• Probability of rolling exactly one six is 3(25/216)

≈ 0.347. • Binomial Probability Formula (n = 3, p = 1/6, q = 5/6

and x = 1). Probability of rolling exactly one 6 is:

xnxxnxxn qp

xxn

nqpCxP

!)!(

!)(

Binomial Probability Formula

Example: Binomial Probabilities

347.072

25

)216

25(3

)36

25)(

6

1(3

)6

5)(

6

1(3

)6

5()

6

1(

!1)!13(

!3)1(

2

131

P

Binomial Probability Distribution

• By listing the possible values of x with

the corresponding probability of each,

we can construct a Binomial

Probability Distribution.

Constructing a Binomial Distribution

In a survey, a company asked their workers and retirees to name their expected sources of retirement income. Seven workers who participated in the survey were asked whether they expect to rely on Pension for retirement income. 36% of the workers responded that they rely on Pension only. Create a binomial probability distribution.

Constructing a Binomial Distribution

044.0)64.0()36.0()0( 7007 CP

173.0)64.0()36.0()1( 6117 CP

292.0)64.0()36.0()2( 5227 CP

274.0)64.0()36.0()3( 4337 CP

154.0)64.0()36.0()4( 3447 CP

052.0)64.0()36.0()5( 2557 CP

010.0)64.0()36.0()6( 1667 CP

001.0)64.0()36.0()7( 0777 CP

x P(x)

0 0.044

1 0.173

2 0.292

3 0.274

4 0.154

5 0.052

6 0.010

7 0.001

P(x) = 1

Notice all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.

Finding a Binomial Probability Using a Table• Fifty percent of working adults spend less than 20 minutes

commuting to their jobs. If you randomly select six working adults, what is the probability that exactly three of them spend less than 20 minutes commuting to work?

• Using the distribution for n = 6 and p = 0.5, we can find the probability that x = 3

Population Parameters of a Binomial Distribution

Mean: = np

Variance: 2 = npq

Standard Deviation: = √npq

Example

• In Murree, 57% of the days in a year are cloudy. Find the mean, variance, and standard deviation for the number of cloudy days during the month of June.

Mean: = np = 30(0.57) = 17.1

Variance: 2 = npq = 30(0.57)(0.43) = 7.353

Standard Deviation: = √npq = √7.353 ≈2.71

Problem 1

Four fair coins are tossed simultaneously.

Find the probability function of the random

variable X = Number of Heads and compute

the probabilities of obtaining no heads,

precisely 1 head, at least 1 head, not more

than 3 heads.

Problem 2

If the Probability of hitting a target in

a single shot is 10% and 10 shots are

fired independently. What is the

probability that the target will be hit at

least once?

Problem 3

If the Probability of hitting a target in

a single shot is 5% and 20 shots are

fired independently. What is the

probability that the target will be hit at

least once?

Problem 5

Let X be the number of cars per minute

passing a certain point of some road between

8 A.M and 10 A.M on a Sunday. Assume that

X has a Poisson distribution with mean 5.

Find the probability of observing 3 or fewer

cars during any given minute.

Problem 7

In 1910, E. Rutherford and H. Geiger

showed experimentally that number of alpha

particles emitted per second in a radioactive

process is random variable X having a

Poisson distribution. If X has mean 0.5. What

is the probability of observing 2 or more

particles during any given second?

Problem 9

Suppose that in the production of 50л

resistors, non-defective items are those that

have a resistance between 45л and 55л and

the probability of being defective is 0.2%. The

resistors are sold in a lot of 100, with the

guarantee that all resistors are non-defective.

What is the probability that a given lot will

violate this guarantee?

Problem 11

Let P = 1% be the probability that a

certain type of light bulb will fail in 24

hours test. Find the probability that a sign

consisting of 100 such bulbs will burn 24

hours with no bulb failures.

Problem 13

Suppose that a test for extrasensory

perception consists of naming (in any

order) 3 card randomly drawn from a

deck of 13 cards. Find the probability that

by chance alone, the person will correctly

name (a) no cards, (b) 1 Card, (c) 2

Cards, and (d) 3 cards.

Ex. 6: Finding Binomial Probabilities

• A survey indicates that 41% of American women consider reading as their favorite leisure time activity. You randomly select four women and ask them if reading is their favorite leisure-time activity. Find the probability that (1) exactly two of them respond yes, (2) at least two of them respond yes, and (3) fewer than two of them respond yes.

Ex. 6: Finding Binomial Probabilities

• #1--Using n = 4, p = 0.41, q = 0.59 and x =2, the probability that exactly two women will respond yes is:

35109366.)3481)(.1681(.6

)3481)(.1681(.4

24

)59.0()41.0(!2)!24(

!4

)59.0()41.0()2(

242

24224

CP

Calculator or look it up on pg. A10

Ex. 6: Finding Binomial Probabilities

• #2--To find the probability that at least two women will respond yes, you can find the sum of P(2), P(3), and P(4). Using n = 4, p = 0.41, q = 0.59 and x =2, the probability that at least two women will respond yes is:

028258.0)59.0()41.0()4(

162653.0)59.0()41.0()3(

351093.)59.0()41.0()2(

44444

34334

24224

CP

CP

CP

542.0

028258162653.351093.

)4()3()2()2(

PPPxP

Calculator or look it up on pg. A10

Ex. 6: Finding Binomial Probabilities

• #3--To find the probability that fewer than two women will respond yes, you can find the sum of P(0) and P(1). Using n = 4, p = 0.41, q = 0.59 and x =2, the probability that at least two women will respond yes is:

336822.0)59.0()41.0()1(

121174.0)59.0()41.0()0(141

14

04004

CP

CP

458.0

336822.121174..

)1()0()2(

PPxP

Calculator or look it up on pg. A10

Ex. 7: Constructing and Graphing a Binomial Distribution• 65% of American households subscribe to cable TV. You randomly select

six households and ask each if they subscribe to cable TV. Construct a probability distribution for the random variable, x. Then graph the distribution.

Calculator or look it up on pg. A10

075.0)35.0()65.0()6(

244.0)35.0()65.0()5(

328.0)35.0()65.0()4(

235.0)35.0()65.0()3(

095.0)35.0()65.0()2(

020.0)35.0()65.0()1(

002.0)35.0()65.0()0(

66666

56556

46446

36336

26226

16116

06006

CP

CP

CP

CP

CP

CP

CP

Ex. 7: Constructing and Graphing a Binomial Distribution• 65% of American households subscribe to cable TV. You randomly select

six households and ask each if they subscribe to cable TV. Construct a probability distribution for the random variable, x. Then graph the distribution.

Because each probability is a relative frequency, you can graph the probability using a relative frequency histogram as shown on the next slide.

x 0 1 2 3 4 5 6

P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075

Ex. 7: Constructing and Graphing a Binomial Distribution• Then graph the distribution.

x 0 1 2 3 4 5 6

P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5 6

P(x)

Relative Frequency

Households

NOTE: that the histogram is skewed left. The graph of a binomial distribution with p > .05 is skewed left, while the graph of a binomial distribution with p < .05 is skewed right. The graph of a binomial distribution with p = .05 is symmetric.

Mean, Variance and Standard Deviation

• Although you can use the formulas learned in 4.1 for mean, variance and standard deviation of a probability distribution, the properties of a binomial distribution enable you to use much simpler formulas. They are on the next slide.

Quiz # 332 CE(B) – 12 NOV 2012

• Let P = 1% be the probability that a certain type of light bulb will fail in 24 hours test. Find the probability that a sign consisting of 10 such bulbs will burn 24 hours with no bulb failures. (3 Marks)

• Write Probability Distribution Function for Multinomial and Hypergeometric distributions. (2 Marks)