"suppose the fraction of undergraduate students who smoke is 15% and the fraction of graduate students who smoke is 23%. if one-fifth of the college students are graduate students and the rest are undergraduates, what is the probability that a student who smokes is a graduate student? given the information in part (a), is a randomly chosen college student more likely to be a graduate or undergraduate student?
b. repeat part (b) assuming that the student is a smoker."

What is the probability that a student who smokes is a graduate student?
What we want to find is: P(G|S)
P(A) = Probability of A
P(A∩B) = Probability of A and B
P(A|B) = Probability of A, given B
U = Undergrad'
G = Graduate
S = Smoker
We just have to use the formula:
P(G|S) = P(G∩S)/P(S)
The statistics given are independent of one another, i.e. they don't  affect one another, this means:
P(A∩B) = P(A) * P(B)
So:
P(G∩S) = 0.2 * 0.23 = 0.046
P(S) = (0.2 * 0.23) + (0.8 * 0.15)
= 0.046 + 0.12 = 0.166
Therefore:
P(G|S) = 0.046/0.166 = 0.2771 ⇒ 0.28 (28%)

A randomly chosen student is more likely to be an undergrad as they comprise 4/5 or 0.8 or 80% of the student body.
Assuming they are a smoker, a randomly chosen student is still more likely to be an undergrad.
We worked out for the first part the probability a student is a graduate and a smoker and found there is only a 28% chance of randomly selecting a graduate from among the smokers.
This means there is a 72% chance you will select an undergrad.

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