Question

To reduce laboratory​ costs, water samples from four public swimming pools are combined for one test for the presence of bacteria. further testing is done only if the combined sample tests positive. based on past​ results, there is a 0.002 probability of finding bacteria in a public swimming area. find the probability that a combined sample from four public swimming areas will reveal the presence of bacteria. is the probability low enough so that further testing of the individual samples is rarely​ necessary?

Answer 1

An interesting and practical problem.

Assume that 
1. prevalence (p=0.002) is uniform in all pools.
2. no more bacteria is introduced during the process of transporting and combining the samples.

Probability that all four pools test negative
=(1-p)^4
=0.998^4
=0.992024

=>

Probability that at least one pool tests positive
= 1-0.992024
= 0.007976
which is approximately four times that of an individual pool.

Assuming the cost of testing 1 sample is the same as cost of testing the combined sample, which is denoted by C.
The long term cost of testing pooled samples of four pools
=0.992024C+0.007976*(5C)  [5C because all four pools must be tested individually]
=1.031C  as opposed to 4C, a savings of 74%.