Question

A committee of four is selected from a total of 4 freshmen, 5 sophomores, and 6 juniors (obviously without replacement since one person can't simultaneously fill two committee spots). Using combinations, determine the probability that:

a. at least three freshmen.are selected

b. all four selected are of the same class (order of selection doesn't matter)

c. all four selected are NOT of the same class (order of selection doesn't matter)

d. exactly three of the same class are selected (order of selection doesn't matter)

Answer 1

Combination formula for selecting r items from n, nCr = n!/(r! x (n-r)!)

Total number of students = 4 + 5 + 6 = 15

a) P(at least 3 freshmen are selected) = P(3 freshmen are selected) + P(4 freshmen are selected)

= 4C3 x 11 / 15C4 + 4C4/15C4

= 0.0322 + 0.0007

= 0.0329

b) P(all 4 are of same class) = P(4 freshmen) + P(4 sophomores) + P(4 juniors)

= 4C4/15C4 + 5C4/15C4 + 6C4/15C4

= 0.0007 + 0.0037 + 0.0110

= 0.0154

c) P(all 4 are not of same class) = 1 - P(all 4 are of same class)

= 1 - 0.0154

= 0.9846

d) P(exactly 3 of the same class are selected) = P(3 freshmen) + P(3 sophomores) + P(3 juniors)

= 4C3x11C1/15C4 + 5C3x10C1/15C4 + 6C3x9C1/15C4

= 0.0322 + 0.0733 + 0.1319

= 0.2374