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 0.008. (round to three decimal places as needed.)
is the probability low enough so that further testing of the individual samples is rarely necessary?
a. the probability is quite low, indicating that further testing of the individual samples will be a rarely necessary event.
b. the probability is quite high, indicating that further testing is necessary for all of the combined samples.
c. the probability is quite low, indicating that further testing is not necessary for any of the combined samples.
d. the probability is quite high, indicating that further testing of the individual samples will frequently be a necessary event.

Explanation:

Step1: Calculate combined - sample probability

Assume the probability of finding bacteria in one public swimming area is $p = 0.004$. The probability that a combined sample from two independent public swimming areas is positive is calculated using the formula for the probability of the union of two independent events. For two independent events $A$ and $B$ with probabilities $P(A)$ and $P(B)$ respectively, the probability that either $A$ or $B$ (or both) occurs is $P(A\cup B)=P(A)+P(B)-P(A)P(B)$. Since $P(A) = P(B)=0.004$ and they are independent, $P(A\cup B)=0.004 + 0.004-0.004\times0.004=0.008 - 0.000016 = 0.007984\approx0.008$.

Step2: Evaluate the probability

A probability of $0.008$ (or $0.8\%$) is considered quite low. A low - probability event means that it does not occur frequently. In the context of further testing of individual samples, a low probability of a positive combined sample indicates that further testing of individual samples will rarely be necessary.

Answer:

A. The probability is quite low, indicating that further testing of the individual samples will be a rarely necessary event.