Question

to reduce laboratory costs, water samples from two 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 0.004 probability of finding bacteria in a public swimming area. find the probability that a combined sample from two 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? the probability of a positive test result is . (round to three decimal places as needed.)

Explanation:

Step1: Find the probability of no - bacteria in a single sample

The probability of finding bacteria in a public swimming area is $p = 0.004$. So the probability of not finding bacteria in a single sample is $q=1 - p=1 - 0.004 = 0.996$.

Step2: Find the probability of no - bacteria in the combined sample

Since the samples are independent, the probability of no - bacteria in the combined sample of two public swimming areas is $q\times q=0.996\times0.996 = 0.992016$.

Step3: Find the probability of a positive test result

The probability of a positive test result (presence of bacteria in the combined sample) is $1 -$ (probability of no - bacteria in the combined sample). So it is $1-0.992016 = 0.007984\approx0.008$.

Answer:

$0.008$