The first step in solving this problem is to recognize what we are trying to calculate and what
quantities we have been given in the question. We can introduce a Bernoulli random variable and
say that $D=1$ when the person has the disease. We can then introduce a second Bernoulli random variable $T$
and say that $T=1$ when a person gets a positive test result. We then note that these two random
variables are not independent.
With these symbols in place we can now state clearly what it we are trying to calculate. We are
trying to calculate the conditional probability $P(D=1|T=1)$. In addition, the question tells us
that:
$$
P(D=1)=0.001 \qquad \qquad P(T=1|D=1)=0.990 \qquad \qquad P(T=1|D=0)=0.005
$$
From these quantities we can the probability of getting a positive test result, $P(T=1)$ using the
partition theorem as shown below:
$$
P(T=1) = P(T=1|D=1)P(D=1) + P(T=1|D=0)P(D=0) = P(T=1|D=1)P(D=1) + P(T=1|D=0)[1-P(D=1)] = 0.99 \times 0.1 + 0.005 \times ( 1 - 0.001 ) = 0.005985
$$
We can now insert this result into Bayes theorem to get the desired conditional probability.
$$
P(D=1|T=1) = \frac{P(T=1|D=1)P(D=1)}{P(T=1)} = \frac{0.99 \times 0.001}{0.005985} \approx 0.165
$$
