Question

There are only 8 software firms that design a certain kind of software. Of​ these, 3 use Protocol A in their software. A working group of 5 designers​ (no more than one from a​ firm, some firms were not​ represented) was formed to reach an a new protocol. Assume that the working group is chosen without regard to whether a​ designer's firm uses Protocol A.

a) What is the probability that exactly 0 of the designers on the working group will be from firms that use Protocol​ A? ​ (Use four decimal​ places.) 

b)What is the mean number of designers on the working group that are from firms that use Protocol​ A? ​ (Use one decimal place in your​ answer.)

c)What is the variance of the number of designers on the working group that are from firms that use Protocol​ A? ​ (Use two decimal places in your​ answer.)  

Answer 1

Let X be the random variable which indicates the number of designers from firms that use Protocol A, among 5 (the working group)

X can take the values 0,1,2,3

The total number of ways of choosing 5 designers from 8 firms is

The total number of ways of choosing X engineers from 3 firms that use Protocol A is

The total number of ways of choosing the remaining 5-X engineers from 5 firms that do not use Protocol A is

The probability of X=x is

We can also state X in terms of a named distribution.

Let N=8 indicate the number of software firms, of which M=3 use Protocol A in their software. Let K=5 designers are selected from these N firms (no more than 1 from a firm)X has a hypergeometric distribution given by the probability

a) The probability that exactly 0 of the designers on the working group will be from firms that use Protocol​ A is

ans: The probability that exactly 0 of the designers on the working group will be from firms that use Protocol​ A is 0.0179

b) We can either use the known formula for expectation of a HyperGeometric distribution, or we can calculate the probabilities for X=0,1,2,3 and use the formula for expectation.

The expectation of a hypergeometric RV is

Alternatively,

We could get the pmf of X (by using the formula above) as

X	P(x)
0	0.0179
1	0.2679
2	0.5357
3	0.1786

the expectation of X is

ans: the mean number of designers on the working group that are from firms that use Protocol​ A is 1.875

c) Using the formula for the variance of Hypergeometric distribution, the variance is

However, using the pmf we get

First get the expectation of

the Variance of X is

ans: the variance of the number of designers on the working group that are from firms that use Protocol​ A is 0.50