7 – SL NB 06 S False Positives V1 V3

Well, let’s see. Let’s use Bayes theorem to calculate it. We’ll use the following notation, S will stand for sick, H will stand for healthy, and the plus sign will stand for testing positive. So since one out of every 10,000 people are sick, we get that P of S is 0.0001. Similarly, P of H is 0.9999. Since the test has 99 percent accuracy, both for sick and for healthy patients, we see that P of plus, given S is 0.99, the probability that the sick patient will get correctly diagnosed. And that P of plus given H is 0.01, the probability that the healthy patient will get incorrectly diagnosed as sick. So plugging that into the new formula, we get the probability of being diagnosed as positive when you’re sick is exactly 0.0098, which is less than 1 percent. Really? Less than 1 percent? When the test has 99 percent accuracy? That’s strange, but I guess that’s the answer to the quiz. So, less than 1 percent falls in this category of 0 to 20 percent. I’m still puzzled though, why less than 1 percent If the test is correct 99 percent of the time. Well, let’s explore. Let’s go back to the tree of possibilities. Let’s say we start with 1 million patients, and they have two options, healthy and sick. Now, since 1 out of every 10,000 patients is sick, then from this group of 1 million patients, 100 will be sick and the remaining 999,900 will be healthy. Now let’s remember that for every 100 patients, 99 get correctly diagnosed and one gets incorrectly diagnosed, this happens both for sick and for healthy patients. So, let’s see how many of these patients will get diagnosed positively or negatively. Out of the 100 sick ones, 99 will be correctly diagnosed as positive and one will be incorrectly diagnosed as negative. Now, out of the healthy ones, 1 percent or 9,999 will be incorrectly diagnosed as positive and the remaining 99 percent or 989,901 will be correctly diagnosed as negative. Now let’s really examine these four groups. The first group is the sick people who we will send for more test or treatment. The second is the unlucky sick people that will be sent home with no treatment. The third is a slightly confused healthy people who will be sent for more tests. And the fourth group or the majority is the people who are healthy and were correctly diagnosed healthy and sent home. But now, here’s the thing, we know we tested positively, so we must be among one of these two groups, the sick people who tested positively or the healthy people who tested positively. One group is much larger, it has 9,999 people, whereas the other one has only 99 people. The probability that we’re in this group is much larger than that we’re in this group. As a matter of fact, the probability that we are in the small group is 99 divided by the sum, 99 plus 9,999, which is, you guessed it, 0.0098, which is smaller than 1 percent, this is the probability of being sick if you are diagnosed as positive. But why is the group of healthy people who tested positively so much larger than the group of sick people who tested positively? The reason is because, even though the test only fails 1 percent of the time, that 1 percent is much, much larger than the one out of 10,000 rate of sickness among the population. In other words, in a group of 10,000 healthy people, 1 percent or a 100 of them will get misdiagnosed as sick. On the other hand, in a group of 10,000 people, around one will be sick, this is much less. So if you know you’ve tested positively, you are still more likely to be among the 100 errors than among the ones sick. And how much more likely? Around 100 times, and that’s why our probability of being sick while being diagnosed positively is around 1 percent. This phenomenon is called the False Positive, and it has been a nightmare for the medical world, the legal world and many others. Search False Positives on Google, and you’ll see many cases in which people have been misdiagnosed, misjudged etc. So always be aware of false positives, they are very sneaky.

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