Applied Statistics and Probability for Engineers

3-6 BINOMIAL DISTRIBUTION 75

general distribution, which includes the binomial as a special case, is the multinomial

distribution.

Examples of binomial distributions are shown in Fig. 3-8. For a fixed n, the distribution

becomes more symmetric as pincreases from 0 to 0.5 or decreases from 1 to 0.5. For a fixed

p, the distribution becomes more symmetric as nincreases.

EXAMPLE 3-17 Several examples using the binomial coefficient follow.

EXAMPLE 3-18 Each sample of water has a 10% chance of containing a particular organic pollutant. Assume

that the samples are independent with regard to the presence of the pollutant. Find the proba-

bility that in the next 18 samples, exactly 2 contain the pollutant.

Let X the number of samples that contain the pollutant in the next 18 samples analyzed.

Then Xis a binomial random variable with p 0.1 and n 18.

Therefore,

P 1 X 22 a

18

2

b 1 0.1 221 0.9 216

a

100

4

b 100 ! 34! 96! 4  1100  99  98  (^972)  14  3  22 3,921,225
a
15
10
b 15 ! 310! 5! 4  115  14  13  12  (^112)  15  4  3  22  3003
a
10
3
b 10 ! 33! 7! 4  110  9  (^82)  13  22  120
a
n
x
b
Figure 3-8 Binomial distributions for selected values of nand p.
0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.03
0.06
0.09
0.12
0.15
0.18
x
(a)
012345678910
0
0.1
0.2
0.3
0.4
x
(b)
np
10 0.1
10 0.9
np
20 0.5
f(x
)
f(x
)
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