Water sample from 6 public swimming pools are combined for one test regarding bacteria level.Further testing is done if the combined test fails. Past results have shown that there is a 2% chance of unacceptably high bacteria level. Determine the probability that the combined sample will fail.
To determine the probability that the combined sample will fail, we need to consider the probability of each individual sample passing the test.
Let's assume that the probability of a single sample passing the test is p (in this case, finding the probability of failure would be equivalent to finding the probability of not passing).
Given that there are 6 public swimming pools, we can assume that each sample has the same probability of passing the test. Therefore, the probability of passing a single sample is p.
Since the probability of passing a single sample is p, the probability of not passing (failing) is 1-p.
Based on the given data, we know that the combined test fails if any of the individual samples fail. So, to determine the probability of the combined sample failing, we need to find the probability that at least one sample fails.
To calculate this probability, we can use the complement rule. The complement of the event "at least one sample fails" is the event "all samples pass." Therefore, the probability of the combined sample failing is equal to 1 minus the probability of all samples passing.
In this case, since there are 6 samples and each sample is independent, the probability of all samples passing is (p)^6 because the probability of passing for each sample is p.
So, the probability of the combined sample failing is 1 minus (p)^6.
Now, we are given that in past results, there is a 2% chance of unacceptably high bacteria levels. This means that the probability of a single sample failing is 0.02, or 2%.
Substituting this value into our equation, the probability of the combined sample failing is 1 minus (0.02)^6.
Simplifying this expression, the probability that the combined sample will fail is approximately 0.1194, or 11.94%.