Patients of a clinic are tested for a particular desease. For each patient, the result of the test – ‘infected’/’not infected’ – is correct with the probability 0.8. Suppose that 20% of the patients are infected. What is the probability that a given patient is indeed infected if his/her test result shows ‘infected’? ◦ 0.4 ◦ 0.5 ◦ 0.6 ◦ 0.64 ◦ 0.8
I suppose the answer is 0.8 since that is the probability that a result is accurate.
Writing $+/-$ for "test positive/negative" and $I/J$ for "infected/not infected", then letting $P(I\cap+)=u$ and $P(J\cap-)=v$, we have $$\begin{array} &&I&J\\ +&u&u/4\\ -&v/4&v\end{array}$$ from which we derive the system $$u+v/4=0.2\\ u/4+v=0.8$$ and solving gives $u=0,v=0.8$. Thus $P(I\mid+)=0$, which is not one of the given answers. A number or two must have been transcribed incorrectly.