Finding a probability of a positive test result from a combined water sample

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to find a probability of a positive test result from a combined water sample.  Here's our problem statement: To reduce laboratory costs, water samples from four public swimming pools are combined for one test for the presence of bacteria.  Further testing is done only if the combined sample test positive.  Based on past results, there is a 0.008 probability of finding bacteria in a public swimming area.  Find the probability that a combined sample from four 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?

Part 1

OK, The first part of this problem asks us to find the probability of a positive test result in the combined sample.  The combined sample will test positive if any one of the four samples taken from each of the four public swimming pools tests positive.  So we got to look at the probability of at least one of these swimming pools testing positive.  Well, the probability that one of those swimming pools is going to test positive is going to be 0.008, as it says here in the problem statement.  But we want the probability that we're going to find at least one of those pools.

OK, so the probably that an individual pool test positive is 0.008, [and] we want the probability that the at least one of those swimming pools from all four considered together is going to be a testing positive for bacteria.  So to calculate that, I'm going to bring up my calculator here.  And the first thing I'm going to do is calculate the probability that we're not going to find the bacteria in any one of the swimming pools, which is just the complement of the probability they give us here.  So I subtract that out from one, and that gives me 99.2%. So this is the probability that we're not going to find any bacteria in an individual swimming pool.

So now if I take that and raise it to the fourth power, because I've got four swimming pools, and then I take this and subtract, this is now the probability that I'm going to find bacteria — excuse me.  This is the probability that I'm not going to find bacteria in any of the swimming pools.  So to find that probability that we — that we don't find it, which is what we're actually looking for, the probability of having a positive test result, then I'm going to have to take this and subtract it from one because that's how — that's the probability of at least one.  It's one minus the complement.  And this is the complement.  So if I take that and I make it negative and then add it to one, that's going to be the same thing as subtracting it from one.  And lo and behold, here's the probability of a positive test result.  I'm asked around to three decimal places.  Well done!

Part 2

And now the second part of this problem asks, "Is the probability low enough?"  So the further testing of the individual samples is rarely necessary.  Well, here we've got 3%, which is a pretty low value.  So yeah, I would say that 3% is low enough.  So we don't need to --- rarely need to do any further testing.  So I'm going to go and click the answer option that tells me that.  It should be this one right here.

Whoops, what did I do wrong?  Oh, it says "will not be a rarely necessary event" and it will be a rarely necessary event.  You've got to watch the --- the little details in the words here; that'll trip you up.  Excellent!

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