Section 6.2 The Binomial Probability Distribution

6-1

Recall: A Discrete random variable

can take whole (countable) values

A Continuous random variable

can take on any value between two specified values

Examples:

• Flipping a coin a number of times: the Number of heads observed is a discrete variable.

• A fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds.

Consequently: Two types of Probability Distributions

Discrete Probability Distribution Continuous Probability Distribution

A Discrete Probability Distribution: The Binomial Distribution

Criteria for a Binomial Probability ExperimentCriteria for a Binomial Probability Experiment

An experiment is said to be a binomial experiment if:

1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial.

2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials.

3. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure.

4. The probability of success is fixed for each trial of the experiment.

6-5

Notation Used in the Notation Used in the Binomial Probability DistributionBinomial Probability Distribution

• There are n independent trials of the experiment.

• p denotes the probability of success so 1 – p is the probability of failure.

• X is the binomial random variable that denotes the number of successes in n independent trials of the experiment. So, 0 < X < n.

Which of the following are binomial experiments?

(a)A player rolls a pair of fair die 10 times. The number X of 7’s rolled is recorded.

(b) The 11 largest airlines had an on-time percentage of 84.7% in November, 2001 according to the Air Travel Consumer Report. In order to assess reasons for delays, an official with the FAA randomly selects flights until she finds 10 that were not on time. The number of flights X that need to be selected is recorded.

(c) In a class of 30 students, 55% are female. The instructor randomly selects 4 students. The number X of females selected is recorded.

EXAMPLE Identifying Binomial ExperimentsEXAMPLE Identifying Binomial Experiments

Binomial experiment

Not a binomial experiment – not a fixed number of trials.

Not a binomial experiment – the trials are not independent.6-7

6-8

Computing a Binomial Probability Value

EXAMPLE Using the Binomial Probability Distribution FunctionEXAMPLE Using the Binomial Probability Distribution Function

According to the Experian Automotive, 35% of all car-owning households have three or more cars.

(a)In a random sample of 20 car-owning households, what is the probability that exactly 5 have three or more cars?

(b) In a random sample of 20 car-owning households, what is the probability that less than 4 have three or more cars?

(c) In a random sample of 20 car-owning households, what is the probability that at least 4 have three or more cars?

5 20 520 5(5) (0.35) (1 0.35)

0.1272

P C

( 4) ( 3)

(0) (1) (2) (3)

0.0444

P X P X

P P P P

( 4) 1 ( 3)

1 0.0444

0.9556

P X P X

6-10

Computing the Mean and Standard Deviation of a Binomial Probability Distribution

According to the Experian Automotive, 35% of all car-owning households have three or more cars. In a simple random sample of 400 car-owning households, determine the mean and standard deviation number of car-owning households that will have three or more cars.

EXAMPLE Finding the Mean and Standard Deviation of a Binomial Random Variable

EXAMPLE Finding the Mean and Standard Deviation of a Binomial Random Variable

(400)(0.35)

140

X np

(1 )

(400)(0.35)(1 0.35)

9.54

X np p

6-11

6-12

Constructing Binomial Probability Histograms

(a) Construct a binomial probability histogram with n = 8 and p = 0.15

(b) Construct a binomial probability histogram with n = 8 and p = 0. 5

(c) Construct a binomial probability histogram with n = 8 and p = 0.85

For each histogram, comment on the shape of the distribution.

EXAMPLESEXAMPLES

n=8, p=0.15

x P(x)

0 0.272491

1 0.384693

2 0.237604

3 0.08386

4 0.018499

5 0.002612

6 0.00023

7 1.16E-05

8 2.56E-07

6-14

n=8, p=0.5

x P(x)

0 0.003906

1 0.03125

2 0.109375

3 0.21875

4 0.273438

5 0.21875

6 0.109375

7 0.03125

8 0.003906

6-15

n=8, p=0.85

x P(x)

0 2.56289E-07

1 1.16184E-05

2 0.000230432

3 0.002611567

4 0.018498596

5 0.083860304

6 0.237604195

7 0.384692506

8 0.272490525

6-16

For a Larger number of trials, n=25:

n=50:

n=70:

Demo

For a fixed probability of success, p, as the number of trials n in a binomial experiment increase, the probability distribution of the random variable X becomes bell-shaped.

As a general rule of thumb, if np(1 – p) > 10, then the probability distribution will be approximately bell-shaped.

6-19

According to the Experian Automotive, 35% of all car-owning households have three or more cars. A researcher believes this percentage is higher than the percentage reported by Experian Automotive. He conducts a simple random sample of 400 car-owning households and found that 162 had three or more cars. Is this result unusual ?

EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment

EXAMPLE Using the Mean, Standard Deviation and Empirical Rule to Check for Unusual Results in a Binomial Experiment

(400)(0.35)

140

X np

(1 )

(400)(0.35)(1 0.35)

9.54

X np p

2 140 2(9.54)

120.9X X

2 140 2(9.54)

159.1X X

The result is unusual since 162 > 159.1(more than 2 standard dev. greater than the mean)