Answer to Question #143915 in Statistics and Probability for Vladimyr Lubin

Question

(4.3.23) Testing for a disease can be made more efficient by combining samples. If the samples from five people are combined and the mixture tests​ negative, then all five 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 is0.2 ​, find the probability of a positive result for five samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely​ necessary?
The probability of a positive test result is ?(Round to 3 decimals places)

Answer 1

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 0.8⁵=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 n_ip_i / 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.