Question

A committee of 3 members is selected from a club made up of 15 junior members and 15 senior members. What is the expected number of juniors in the committee? The expected number of juniors in the commitee is (Round to two decimal places as needed.)

Answer 1

Juniors = 15

Seniors = 15

Total members = 30

Number of people selected = 3

Let X = Number of Juniors selected.

X can take values 0, 1, 2, 3

For X = 0:
Number of ways of selecting 3 people from 30 people =

$\binom{30}{3}=4060$

Number of ways of selecting 3 Seniors from 15 Seniors =

$\binom{15}{3}=455$

So,

P(X=0) = 455/4060 = 0.1121

For X = 1:
Number of ways of selecting 3 people from 30 people =

$\binom{30}{3}=4060$

Number of ways of selecting 2 Seniors from 15 Seniors =

$\binom{15}{2}=105$

Number of ways of selecting 1 Junior from 15 Juniors =

$\binom{15}{1}=15$

So,

P(X=1) = 105 X 15 /4060 = 0.3879

For X = 2:
Number of ways of selecting 3 people from 30 people =

$\binom{30}{3}=4060$

Number of ways of selecting 1 Senior from 15 Seniors =

$\binom{15}{1}=15$

Number of ways of selecting 2 Junior2 from 15 Juniors =

$\binom{15}{2}=105$

So,

P(X=2) = 15 X 105 /4060 = 0.3879

For X = 3:
Number of ways of selecting 3 people from 30 people =

$\binom{30}{3}=4060$

Number of ways of selecting 3 Juniors from 15 Juniors =

$\binom{15}{3}=455$

So,

P(X=3) = 455/4060 = 0.1121

Thus, the probability Mass Function of X is give by:

x	p	x p
0	0.1121	0
1	0.3879	0.3879
2	0.3879	0.7758
3	0.1121	0.3363
Total =		1.50

So,

Expected Value of Number of Juniors in the Committee = 1.50

So,

Answer is:

1.50