There is a group of 15 students (3 seniors, 5 juniors, and 7 sophomores). Suppos
ID: 3301878 • Letter: T
Question
There is a group of 15 students (3 seniors, 5 juniors, and 7 sophomores). Suppose that 4 students are randomly selected from this group.
a) What is the probability that exactly 2 of the selected students are sophomores?
b) What is the probability that all 4 of the selected students will be in the same category?
c) What is the probability that at least one student of each type is selected?
d) Suppose that students are to be selected one by one until a junior student is found. What is the probability that is it necessary to examine at least six students?
Explanation / Answer
a) P(exactly 2 students are sophomores) = 7C2 * 8C2 / 15C4
= 21 * 28 / 1365
= 0.4308
b) P(all 4 from same category) = P(all seniors) + P(all juniors) + P(all sophomores)
= 0 + 5C4/15C4 + 7C4/15C4
= 5/1365 + 35/1365
= 0.0293
c) P(at least one student of each type) = P(2 seniors, 1 junior and 1 sophomore) + P(1 senior, 2 juniors and 1 sophomore) + P(1 senior, 1 junior and 2 sophomores)
= 3C2 * 5C1 * 7C1 / 15C4 + 3C1 * 5C2 * 7C1 / 15C4 + 3C1 * 5C1 * 7C2 / 15C4
= 3*5*7/1365 + 3*10*7/1365 + 3*5*21/1365
= 0.4615
d) P(examine at least 6 students) = 1 - [P(examine 1 student) + P(examine 2 students) + P(examine 3 students) + P(examine 4 students) + P(examine 5 students)]
= 1 - [3/15 + 12/15 * 3/14 + 12/15 * 11/14 * 3/13 + 12/15 * 11/14 * 10/13 * 3/12 + 12/15 * 11/14 * 10/13 * 9/12 * 3/11]
= 0.2637
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