[Math] how to approach this sort of probability problem

Question

Question 2: Suppose a new Internet company Mumble.com was to require
all employees to take a drug test. Mumble.com can aord only the inexpensive
drug test { the one with a 5% false positive and a 10% false negative
rate. (That means that ve percent of those who are not using drugs will incorrectly
test positive, and ten percent of those who are actually using drugs
will test negative.) Suppose that 10% of those who work for Mumble.com are
using the drugs for which Mumble.com is checking. An employee is chosen
at random.
(a) (3 marks). What is the probability the employee both uses drugs and
tests positive?
(b) (2 marks). What is the probability the employee does not use drugs
but tests positive anyway?
(c) (2 marks). What is the probability the employee tests positive?
(d) (2 marks). If the employee has tested positive, what is the probability
he or she uses drugs?

Answer 1

(a) The probability she uses drugs is $0.1$. Given she uses drugs, the probability she tests positive is $0.9$. Multiply.

(b) Same idea, a little simpler.

(c) Add the answers to (a) and (b).

(d) Let $D$ be the event she uses drugs, and $P$ be the event she tests positive. We want $\Pr(D|P)$. By the defining formula for conditional probability, we have $$\Pr(D|P)=\frac{\Pr(D\cap P)}{\Pr(P)}.$$ The numerator has been calculated in (a), and the denominator in (c).