MathProblem 34 Drug test problem

10% of the people in a certain population use an illegal drug. A drug test yields the correct result 90% of the time, whether the person uses drugs or not. A random person is forced to take the drug test and the result is positive. What is the probability he uses drugs?

Solution

利用全概率公式：

\[P(use|positive) = \frac{P(positive|use)\cdot P(use)}{P(positive|use)\cdot P(use)+P(positive|unuse)\cdot P(unuse)}\\ =\frac{0.9\times 0.1}{0.9\times 0.1+0.1\times 0.9}=\frac{1}{2} \]