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Unit 14: Theoretical Probability Distributions

Let the random variable r denote the number of defective televisions. In terms of notations, we Notes
can write N = 10, k = 4 and n = 3.

4 C ´ C
6

P
Thus, we can write ( ) r = r 3 r , r = 0,1,2,3
10
C
3
The distribution of r is hypergeometric. This distribution can also be written in a tabular form
as given below :
r 0 1 2 3 Total
5 15 9 1
P ( ) r 1
30 30 30 30

14.2.1 Binomial Approximation to Hypergeometric Distribution

In sampling problems, where sample size n (total number of trials) is less than 5% of population
size N, i.e., n < 0.05N, the use of binomial distribution will also give satisfactory results. The
reason for this is that the smaller the sample size relative to population size, the greater will be
the validity of the requirements of independent trials and the constancy of p.

Example 12: There are 200 identical radios out of which 80 are defective. If 5 radios are
selected at random, construct the probability distribution of the number of defective radios by
using (i) hypergeometric distribution and (ii) binomial distribution.

Solution.
(i) It is given that N = 200, k = 80 and n = 5.
Let r be a hypergeometric random variable which denotes the number of defective radios,
then
80 120
C ´ C 5 r

r
P ( ) r = , r = 0,1,2,3,4,5
200
C
5
The probabilities for various values of r are given in the following table :
r 0 1 2 3 4 5 Total
P ( ) 0.0752 0.2592 0.3500 0.2313 0.0748 0.0095r 1

(ii) To use binomial distribution, we find p = 80/200 = 0.4.

r
5

P ( ) r = C r (0.4 ) (0.6 ) 5 r , r = 0,1,2,3,4,5
The probabilities for various values of r are given in the following table :
r 0 1 2 3 4 5 Total
P ( ) 0.0778 0.2592 0.3456 0.2304 0.0768 0.0102r 1

We note that these probabilities are in close conformity with the hypergeometric
probabilities.

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