Question

A professor of a law school observes that only 25% of students who get admitted to the freshman class reach fourth year. assuming that this is correct, among 15 randomly selected first year students,
a. find the probability that exactly 8 will reach fourth year.
b. find the probability that at most 7 will reach fourth year.
c. find the probability that between 7 and 10, inclusive will reach fourth year.
d. find the expected number of students who will reach fourth year for the next 25 students.

Answer 1

Question #14649A professor of a law school observes that only 25% of students who get admitted to the freshman class reach fourth year. assuming that this is correct, among 15 randomly selected first year students, a. find the probability that exactly 8 will reach fourth year.

b. find the probability that at most 7 will reach fourth year.

c. find the probability that between 7 and 10, inclusive will reach fourth year.

d. find the expected number of students who will reach fourth year for the next 25 students. .

Solution. The condition implies that the probability for a particular student to reach the fourth course in 0.25. Next, let $\xi$ be the random variable, that equals number of students out of 15 freshmen that reach the fourth course. Then it has binomial distribution $Bin(15,0.25)$. Thus,

a) $P(\xi=k)={15\choose 8}0.25^{8}0.75^{7}\approx 0.013$.

b) $P(\xi\leq 7)=\sum_{k=0}^{7}{15\choose k}0.25^{k}0.75^{15-k}\approx 0.9827$.

c) $P(7\leq\xi\leq 10)\approx 0.056$.

d) $E\xi=15\cdot 0.25\approx 3.75$.

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