Binomial distribution

Nutan S. MishraDepartment of Mathematics and

StatisticsUniversity of South Alabama

Binomial experimentAn experiment is called binomial experiment

if it satisfies following four conditions1. Consists of n trials 2. All the trials are independent.3. There are only two possible outcomes of

each trial.4. Probability of success in each trial is

constant say p.X is number of successes in the experiment

ExampleToss four fair coins.

X= # heads showed up

Then x may take values 0 or 1 or 2 or 3 or at the most 4.

p = .5, n =4 . Use binomial table to complete the following table

x P(X) = 4Cx px(1-p)4-x

0 4C0 p0(1-p)4 =

1 4C1 p1(1-p)3 =

2 4C2 p2(1-p)2 =

3 4C3 p3(1-p)1 =

4 4C4 p4(1-p)0 =

ExampleIn a company it is known that among the

population of employees 35% are smokers and 65% are non smokers.

If we select a sample of size 10 employees from this population and count the number of smokers is the sample then

X= # smokers in the sample of 10 is a binomial variable with n=10 and p= .35 that is (1-p) = .65.

The possible values x takes : 0 to 10Use table of combinations and a calculator to complete the table on next slide

X

# smokers in the sample of size10

P(X)= 10Cx px(1-p)10-x

010C0 p0(1-p)10 =

110C1 p1(1-p)9=

210C2 p2(1-p)8 =

310C3 p3(1-p)7=

410C4 p4(1-p)6=

510C5 p5(1-p)5=

610C6 p6(1-p)4=

710C7 p7(1-p)3=

810C8 p8(1-p)2=

910C9 p9(1-p)1=

1010C10 p10(1-p)0=