According to the record of the registrar's office at a state university, 40% of the students...

According to the record of the registrar's office at a state university, 40% of the students are freshmen, 30% are sophomores, 20% are juniors, and the rest are seniors. Among the freshmen, sophomores, juniors, and seniors, the portion of students who live in the dormitory are, respectively, 75%, 50%, 35%, and 15%.

If a randomly selected student does not live in the dormitory, what is the probability that the student is a freshman?

Expert Solution

P(freshmen) = 0.4

P(sophomores) = 0.3

P(juniors) = 0.2

P(seniors) = 1 - (0.4 + 0.3 + 0.2) = 0.1

P(live in dormitory | freshmen) = 0.75

P(live in dormitory | sophomores) = 0.5

P(live in dormitory | juniors) = 0.35

P(live in dormitory | seniors) = 0.15

P(live in dormitory) = P(live in dormitory | freshmen) * P(freshmen) + P(live in dormitory | sophomores) * P(sophomores) + P(live in dormitory | juniors) * P(juniors) + P(live in dormitory | seniors) * P(seniors)

= 0.75 * 0.4 + 0.5 * 0.3 + 0.35 * 0.2 + 0.15 * 0.1

= 0.535

P(doen't live in dormitory | freshmen) = 1 - P(live in dormitory | freshmen) = 1 - 0.75 = 0.25

P(freshmenn | doen't live in dormitory) = P(doen't live in dormitory | freshmen) * P(freshmen) / P(doen't live in dormitory)

= 0.25 * 0.4 / (1 - 0.535)

= 0.215 (ans)

orchestra answered 2 years ago

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