Solutions to Probability Essentials - Chapter 3

Probability

Problem 3.6 Denoted blood is screened for AIDS. Suppose the test has 99% accuracy, and that one in ten thousand people in your age group are HIV positive. The test has a 5% false positive rating, as well. Suppose the test screens you as positive. What is the probability you have AIDS? Is it 99%?

Solution: E_1="test positive", E_2="test negative". A_1="You have AIDS", A_2="You don't have AIDS". Now we know $P(E_1|A_1)=99\%$, we need to find $P(A_1|E_1)$. Since "one in ten thousand people in your age group are HIV positive", $P(A_1)=1/10000$."5% false positive rating" means $P(E_1|A_2)=5\%$. By Bayes' Theorem

$$\begin{eqnarray*} P(A_1|E_1) &=& \frac{P(E_1|A_1)P(A_1)}{P(E_1|A_1)P(A_1)+P(E_1|A_2)P(A_2)} \\ &=& \frac{99\%\times\frac{1}{10000}}{99\%\times \frac{1}{10000}+5\%\times\frac{9999}{10000}} \\ &\approx& 0.198\% \end{eqnarray*}$$
Note: 关于这个问题的讨论请戳这里：《概率论札记-2-用贝叶斯定理来讨论“医疗诊断的可靠性到底有多少”》