Question

Assume that the class consists of 40 percent freshmen, 20 percent sophomores, 20 percent juniors, and 20 percent seniors. Assume further that 40 percent of the freshmen, 45 percent of the sophomores, 40 percent of the juniors, and 20 percent of the seniors plan to go to medical school. One student is selected at random from the class. (1) What is the probability that the student plans to go to medical school? equation editorEquation Editor (2) If the student plans to go to medical school, what is the probability that she is a freshman?

Answer 1

Here there are given and we assume that Freshman is F, Sophomores is S, Juniors is J, Seniors is SE

Pr(Freshman) = Pr(F) = 40% = 0.4

Pr(Sophomores) = Pr(S) = 20% = 0.20

Pr(Juniors) = Pr(J) = 20% = 0.20

Pr(Seniors) = Pr(SE) = 20% = 0.20

Now further given, and if assume Going to medical school is event M

Pr(Going to medical school if freshman) = Pr(M l F) = 0.40

Pr(Going to medical school if sophomores) = Pr(M l S) = 0.45

Pr(Going to medical school if Juniors) = Pr(M l J) = 0.40

Pr(Going to medical school if Seniors) = Pr(M l SE) = 0.20

Now we have to find

(1) What is the probability that the student plans to go to medical school?

so here we have to find Pr(M)

so we will calculate it by total probability theorem

Pr(M) = Pr(M l F) * Pr(F) + Pr(M l S) * Pr(S) + Pr(M l J) * Pr(J) + Pr(M l SE) * Pr(SE)

= 0.40 * 0.40 + 0.45 * 0.20 + 0.40 * 0.20 + 0.20 * 0.20 = 0.37

(2) If the student plans to go to medical school, what is the probability that she is a freshman?

Or writing it mathematically,

we have to find Pr(F l M)

by bayes theorm we will evaluate it

Pr(F l M) = Pr(M l F) * Pr(F)/Pr(M) = 0.40 * 0.40/0.37 = 0.4324