Question

to reduce laboratory costs, water samples from three 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 0.005 probability of finding bacteria in a public swimming area. find the probability that a combined sample from three 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 probability of no bacteria in a single sample

The probability of finding bacteria in a public swimming area is $p = 0.005$. So the probability of not finding bacteria in a single sample is $q=1 - p=1 - 0.005 = 0.995$.

Step2: Find probability of no bacteria in all three combined samples

Since the samples are independent, the probability that none of the three samples have bacteria is $q\times q\times q=0.995\times0.995\times0.995 = 0.995^{3}$.

Step3: Find probability of positive test result (presence of bacteria in combined sample)

The probability of a positive test result (presence of bacteria in the combined sample) is $P(X\geq1)=1 - P(X = 0)$. Here $P(X = 0)=0.995^{3}$, so $P(X\geq1)=1-0.995^{3}=1 - 0.985074875=0.014925125\approx0.015$.

Answer:

$0.015$