\[ P(Disease | Positive test) = \frac{P(Positive test | Disease) P(Disease)}{P(Positive test)} \]

Now, we already know $P(Positive test | Disease)$ and $P(Disease)$. We can also observe that

14bb19202ba49e3eb1ef942042b9b0df.png

(i.e. we have to consider both true positives and false positives, and the relative probability of each, in working out the overall probability of a positive result), and thus

\[ P(Positive test) = (0.001 \times 0.95) + (0.999 \times 0.10) = 0.101 \]

Thus, we can substitute these known probabilities into Bayes’ Theorem to find out $P(Disease | Positive test)$:

\[ P(Disease | Positive test) = \frac{0.95 \times 0.001}{0.101} = 0.0094 \]