 So let's talk about testing. For better or worse, modern society is based around testing. For example, a medical test. We tend to approve of these because early treatment saves lives. There are professional certification tests. We like it when the people we employ are actually competent. Drug test. You probably want to know whether the pilot of your airplane is on opiates, whether it's so important that the recipient of food stamps be drug-free is not quite as clear. And loyalty tests, which don't exist in a free society because if they did, we wouldn't have a free society. Because testing has such profound real-world consequences, we have to ask the important question, how much should we rely on test results? So imagine we have a tester that's capable of examining a subject, and it'll mark them red if they have a condition, and green if they don't. But no test is perfect. And if they examine a large number of subjects with the condition, they'll make a mistake every now and then. And every now and then, someone with the condition is marked green. Similarly, even though they should be marking everyone without the condition green, they will, from time to time, make a mistake, and mark someone without the condition red. What this means is that there are four possibilities for the results of our test. First, a person could have been marked as having the condition. We call this a positive result. Now, if they are marked as having the condition and they actually have it, we say the result is a true positive. On the other hand, they might have been mismarked. They might have been marked as having the condition, but they don't actually have it. And this produces what's called a false positive result. Another set of possibilities is that a person could have been marked as not having the condition, a negative result. If they do not, in fact, have the condition, this is called a true negative. But if an error was made and they're marked as not having the condition even when they do, this is a false negative. So here's the important question. We don't know whether a person actually has the condition. We're relying on the test result to tell us that. If a test indicates that a person has a condition, do they in fact have the condition? To answer this question, we need to know a little bit more about the test. The sensitivity of the test is how often it correctly indicates the presence of a condition. So if we apply our test to 10 people with the condition, and the test marks 9 of them as having the condition, we say that the test is 90% sensitive. A related idea is the specificity of the test. The specificity of the test is how often it correctly indicates the absence of a condition. So if we apply our test to 10 people without the condition, and the test marks 9 of them as free of the condition, the test is 90% specific. So we might think that a test that's 90% sensitive and 90% specific is going to be a reasonably accurate test. But in fact, that's going to depend on how common the condition is in our population. So suppose we're looking at something that affects 1 in 10 people. Now there's a simple mathematical formula that will allow us to calculate the answer. But if I ran that computation, you wouldn't believe it. How do I know that? Because I didn't believe it myself the first time I ran across this problem. I checked my computations three times looking for the mistake that I had made because I didn't believe the answer that I got. And only after I couldn't find the mistake, I eventually ran an example, just like the one I'm about to show you, to convince myself that I hadn't made a mistake. So let's take a look at that. Since the condition is present in 1 out of 10 people, then out of every group of 10 people, we'd expect one person to have the condition. Suppose we test 100 people. Our tester will go over these and mark these people positive or negative, red or green, with the occasional mistake due to the specificity of 90% and the sensitivity of 90%. Now let's look at those people who tested positive, all of those people who got marked red by the tester. Of these people, what fraction actually have the condition? The surprising answer? Only half of those people who tested positive actually have the condition. Now you might suspect that this is a result of some high-tech smoke and mirrors. To see what happened, we can tabulate our data. So to start off with, we have a total of 100 people that we're testing, and since 1 in 10 of them have the condition, then we know that 10 people have the condition and 90 people do not have the condition. Now because our test is 90% sensitive, then 9 out of 10 of the people with the condition will test positive and one person will test negative. And because our test is 90% specific, then 90% of those without the condition will test negative. That's 81 people, but that also means that 9 people who do not have the condition will test positive. These are the false positive results, and altogether that means that 9 plus 9, 18 people will test positive, but only half of them will actually have the condition. And here's the thing to recognize. The test actually gives us the correct result in the 9 positive results for the people who have the condition and the 81 negative results for the people who don't have the condition. So the test gave the correct result in 9 plus 81, 90 cases, so it's a 90% accurate test. But fully half of those who tested positive do not have the condition. And what this means is that if we're going to make life-altering decisions on the basis of such a test, we need to be very careful about what we do with positive test results. 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