Question

A special committee of five professionals is formed from a group of four lawyers, three doctors and three teachers. At least three lawyers are needed in the special committee. Define X as the number of teachers in the special committee.
a) Construct the probability distribution of the random variable X.
b) Calculate the expected of X and provide an interpretation of it.
c) Calculate the standard deviation of X

Answer 1

a)

Total number of peoples from which we need to select five professionals: 4+3+3 = 10.

Number of ways of selecting at least three lawyers is:

C(4,3) * C(7,2) = 84

Since at least three lawyers are needed so X can take values 2, 1, and 0.

When X=2, means 3 lawyers and 2 teachers are selected. So number of ways of doing this is

C(4,3)*C(3,2)*(3,0) = 12

When X=1, means 3 lawyers and 1 teacher are selected. And rest one professional can be one of remaining 1+3 =4 doctors and lawyers. So number of ways of doing this is

C(4,3)*C(3,1)*(4,1) = 48

When X=0, means 3 lawyers and 0 teacher are selected. And rest two professionals can be selected from remaining 1+3 =4 doctors and lawyers. So number of ways of doing this is

C(4,3)*C(3,0)*(4,2) = 24

Therefore the probability distribution of X is:

X	P(X=x)
2	12/84 = 1/7
1	48/84 = 4/7
0	24/84 = 3/7

(b)

Following table shows the calculations for expected value:

X	P(X=x)	xP(X=x)
2	1/7	2/7
1	4/7	4/7
0	3/7	0
Total		6/7

That is

The expected number of teachers in the commitee is 6/7.

(c)

Following table shows the calculations for standard deviation:

X	P(X=x)	xP(X=x)	x^2P(X=x)
2	1/7	2/7	4/7
1	4/7	4/7	4/7
0	3/7	0	0
Total		6/7	8/7

The standard deviation is: