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Binomial distribution
Nutan S. Mishra
Department of Mathematics and
Statistics
University of South Alabama
Binomial experiment
An experiment is called binomial experiment
if it satisfies following four conditions
1. 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
Example
Toss 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
0
1
2
3
4
P(X) = 4Cx px(1-p)4-x
0(1-p)4 =
C
p
4 0
1(1-p)3 =
C
p
4 1
2(1-p)2 =
C
p
4 2
3(1-p)1 =
C
p
4 3
4
0
4C4 p (1-p) =
Example
In 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 10
Use 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
0
0(1-p)10 =
C
p
10 0
1
1(1-p)9=
C
p
10 1
2(1-p)8 =
C
p
10 2
3(1-p)7=
C
p
10 3
4(1-p)6=
C
p
10 4
2
3
4
5
6
7
8
9
10
5(1-p)5=
C
p
10 5
6(1-p)4=
C
p
10 6
7(1-p)3=
C
p
10 7
8(1-p)2=
C
p
10 8
9(1-p)1=
C
p
10 9
10(1-p)0=
C
p
10 10