Question

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

Complete Question

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Answer:

The  probability is  $P(X \ge 1 ) = 0.003996$

Given that the probability is less than 0.05 , it means that it is  low enough so that further testing of the individual samples is rarely​ necessary

Step-by-step explanation:

From the question we are told that

     The probability of finding bacteria in a public swimming area  is  p =  0.002

 Generally the probability of not finding bacteria in a public swimming area  is mathematically represented as

      $q = 1- p$

=>    $q = 1- 0.002$

=>   $q =0.998$

Generally the the probability that a combined sample from public swimming areas will reveal the presence of bacteria is equivalent to the probability that at least one of the sample will reveal the  presence of bacteria and this is mathematically represented as

       $P(X \ge 1 ) = 1 - P(X')$

Here  $P(X')$ means that none of the 2  sample have bacteria which is mathematically represented as

       $P(X') = q^2$

Here 2 signifies that we are considering two samples  

So  

        $P(X') = 0.998^2$

        $P(X') = 0.996004$

So

       $P(X \ge 1 ) = 1 - 0.996004$

=>     $P(X \ge 1 ) = 0.003996$