Probability of having a disease

Question

5% of the population has a disease. A test gives the correct diagnosis (+ or -) 99% of the time. What is the probability that a person will test positive? If a person does test positive, what is the probability they have it?

I'm not sure how to divide up the probability space.

Let $D = {\text{has disease}}$

Let $+ = {\text{test positive}}, - = {\text{test negative}}$

$P[D] = .05$

$P[\text{correct}] = P[+|D]P[D] + P[-|D^C]P[D^C] = .99$

$P[+] = P[+|D]P[D] + P[+|D^C]P[D^C]$

Is this correct? I'm not sure how to proceed.

Answer 1

You want $P(D|+) = \frac{P(+|D)P(D)}{P(+)}=\frac{P(+|D)P(D)}{P(+|D)P(D) + P(+|\sim D)P(\sim D)}$