A condition affects 1% of people. A test detects it in 90% of those who have it, and wrongly shows it in 10% of those who do not. A person tests positive. What is the probability that they have the condition?

Explanation

1. 01

   Let $D$ be having the condition and $T$ a positive test: $P(T) = 0.01 \times 0.9 + 0.99 \times 0.1 = 0.009 + 0.099 = 0.108$.

2. 02

   $P(D \mid T) = \frac{0.009}{0.108} = \frac{1}{12}$.