# Conditional probability (false positives and negatives) question. What Benefit-cost ratio is required to continue with testing program

The full question can be viewed here - https://imgur.com/a/Vrd47ng

I understand how to answer all but the very last part, (D). I'll include a summary of the question below.

The problem is a very classic one in conditional probability.

QUESTION

5% of the population have the disease (D). A test is available that has a 10% false positive and a 10% false negative rate.

part (B) of the question asks what is the probability of having the disease given that you test positive. I'm quite confident the answer to this is ~ 32%

Part (D) - Suppose that there is no cost or benefit from testing negative but a benefit B from true positives which detect the disease and a cost C to false positives.

i) What is the expected value of the testing programme?

ii) What benefit-cost ratio would you require to proceed with the testing programme?

I'm not quite sure to how to answer either of part D. My somewhat educated guesses are

the expected value = 0.045B - 0.095C

and the benefit to cost ratio required to proceed with the testing programme is 2.11 (or greater)

Thanks in advance for your help.

## 1 Answer

Your answers are not only educated but also correct.

The expected value per person is

\begin{eqnarray*} \mathsf P(TP\cap D)B-\mathsf P(TP\cap ND)C &=&\mathsf P(TP\mid D)\mathrm P(D)B-\mathsf P(TP\mid ND)\mathrm P(ND)C \\ &=& 0.9\cdot0.05\cdot B-0.1\cdot0.95\cdot C \\ &=& 0.045B-0.095C \;. \end{eqnarray*}

• thanks for the confirmation. Any idea about part ii). My thought process here was that expected value should be positive to justify continuing with the testing programme. So if 0.045B-0.095C > 0, then B > 2.11C. Would this approach be correct for what is asked? – NumberCruncher Aug 18 '18 at 12:06
• @NumberCruncher: Yes, that's exactly right. – joriki Aug 18 '18 at 20:52