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In: Statistics and Probability

To reduce laboratory​ costs, water samples from five public swimming pools are combined for one test...

To reduce laboratory​ costs, water samples from five 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 0.007 probability of finding bacteria in a public swimming area. Find the probability that a combined sample from five 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.)

Solutions

Expert Solution

SOLUTION:

From given data,

Further testing is done only if the combined sample tests positive. Based on past​ results, there is a 0.007 probability of finding bacteria in a public swimming area. Find the probability that a combined sample from five 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.

From the information, observe that the number of public swimming pools is 2.

These swimming pools are combined for one test for the presence of bacteria.

Based on the further results, there is 0.007 probability of Ending the bacteria in a public swimming area.

Calculate the probability that the combined sample from two public swimming areas will reveal the presence of bacteria.

The probability of not Ending bacteria in public swimming pool area is 0.993 (= 1 - 0.007)

Hence, the combined sample from five public swimming areas will reveal the presence of bacteria is obtained by considering at least one in a public swimming area.

That is,


P(Combined sample with bacteria ) = P (Atleast one public swimming area  with bacteria )

=1- P (None out of 2 has bacteria)

=1- (0.993 )2

= 1 - 0.986049

= 0.013

Therefore, the probability that a combined sample from two public swimming areas will
reveal the presence of bacteria is

Yes. Here, the probability that the combined sample reveals a presence of bacteria is 0.013, which is less than 0.05

Hence, the probability is low enough so that further testing of the individual samples is necessary.

orchestra answered 1 year ago
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