Bayes' Rule

Question

At a gathering of undergraduate students, 16% were freshmen, 29% sophomores, 42% juniors, and 13% seniors. The percentage who had 8 o'clock classes were the following: freshmen, 37%; sophomores, 25%; juniors, 19%; and seniors, 4%. Find the percentage of those that have 8 o'clock classes that are sophomores. (Round your answer to one decimal place.)

Answer 1

We're interested in P(sophomore|8:00 class). Use Bayes' Theorem:

 

P(S|8) = (P(8|S)P(S)) / P(8)

 

 

 

P(8|S) is given to you in the problem. It's 0.25.

 

P(S) is given to you in the problem. It's 0.29.

 

P(8) requires that you figure out the percentage of students present with 8:00 classes. We'll figure it out year by year, then add the percentages together. There are 0.16(0.37) = 0.0592 freshmen with 8:00 classes, 0.29(0.25) = 0.0725 sophomores with 8:00 classes, 0.42(0.19) = 0.0798 juniors with 8:00 classes, and 0.13(0.04) = 0.0052 seniors with 8:00 classes. That's a total of 0.0592+0.0725+0.0798+0.0052 = 0.2167.

 

 

 

Plug those numbers in:

 

P(S|8) = (P(8|S)P(S)) / P(8)

P(S|8) = ((0.25)(0.29)) / 0.2167

P(S|8) = 0.0725/0.2167

P(S|8) = 0.335

 

The percentage of those who have 8:00 classes that are sophomores is 33.5%.