Testing for a disease can be made more efficient by combining samples. If the samples from people are combined and the mixture tests negative, then samples are negative. On the other hand, one positive sample will always test positive, no matter how many negative samples it is mixed with. Assuming the probability of a single sample testing positive is , find the probability of a positive result for samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely necessary?

The probability of testing positive for one is 0.20.

The probability of testing negative for one sample is (1-0.2)=0.8.

We only save time when all five are negative, which has a probability of 0.8^5=0.32768.

This means that the expected number of tests is

combined sample tests negative = 1 with probability 0.32768

combined sample tests positive = 1+5 retests = 6 with probability 0.67232

Expected number of tests

=Σ nipi / n

=(1*0.32768+6*0.67232)/5 [divide by 5 because we tested 5 samples]

= 0.87232 < 1

So yes, there is a saving.

Note: In practice, all medical tests are not absolute, i.e. they give false-positives(α) and false-negatives (β). The ratios 1-α and 1-β are respectively measures of specificity and sensitivity.

These two parameters complicate the simplistic evaluation above.

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