According to the record of the registrar's office at a state university, 35% of the students are freshman, 25% are sophomore, 16% are junior and the rest are senior. Among the freshmen, sophomores, juniors and seniors, the portion of students who live in the dormitory are, respectively, 80%, 60%, 30% and 20% If a randomly selected student lives in the dormitory, what is the probability that the student is a freshman (tip: Bayes theorem)? a. 0.331 b 0.439 c. 0.638 : d. 0.532

The probability that the student is a freshman is 52.32% If a randomly selected student lives in the dormitory. The right answer is d.

Explanation:

According to the data we have the following:

p(freshman) = 0.35

p(sophomore) = 0.25

p(Junior) = 0.16

p(senior) = 0.24

Also:

p(Dormitory/freshman) = 0.8

p(Dormitory/sophomore) = 0.6

p(Dormitory/Junior) = 0.3

p(Dormitory/Senior) = 0.2

Therefore, to calculate the probability that the student is a freshman If a randomly selected student lives in the dormitory we would habe to use the

Bayes theorem

Hence, p(Freshman/Dormitory) = p(Dormitory/freshman) * p(freshman) / {p(Dormitory/freshman) * p(freshman) +  p(Dormitory/sophomore) * p(sophomore) + p(Dormitory/Junior) * p(Junior) +   p(Dormitory/Senior) * p(senior)}

= 0.8 * 0.35 / { 0.8*0.35 + 0.6*0.25 + 0.3*0.16 + 0.2 * 0.24}

= 0.28/0.526

= 0.532. The probability that the student is a freshman is 52.32%

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