[MUSIC] All right, so let's try this out. So this is a question. Say you know that 1% of women at age 40 who participate in routine screening have breast cancer. And let's say you know that 80% of women who actually do have breast cancer, will get a positive result from the test. Further, say you know that 9.6% of women who do not have breast cancer will also get positive results. So this is the false positive rate, okay. Now given that you know that a woman in this age group had a positive test, all right, the test came back positive in a routine screening, what is the probability that she actually has breast cancer? And you should take a minute to sort of work this out. All right, so what's sort of remarkable is that this is a fairly straightforward application of Bayes' Rule, but intuitively it's easy to make mistakes in the reasoning. And in fact, there's been some famous studies that are now somewhat out of date, where they asked doctors this question. And the doctors came up with wildly wrong answers. Only 15% of the doctors they asked were able to answer the question correctly. So, here's how to break it down. Again with our two by two grid. We've got 1% of cases that have cancer, which means we have 99% of cases that do not have cancer. This is in the global population, and let's say we just know this. All right, we know that 1% of people all across the board have cancer, or at least in this group that we're studying across the board. Now, we also know that, if you have cancer, there's an 80% chance that when you take the test, it will come back positive. Okay, so the true positive rate is 1% have cancer times 80%. Now, we're also given that, if you do not have cancer and you take the test, you'll get a positive result at a rate of 9.6%. And so the false positive rate is the rate of not having cancer, 99% times 9.6%. Now, similarly for the false negative and true negative tests, 1% times 20 times 1 minus 80 and 99% times 1 minus 9.6 is 90.4, okay? So, we have all the information we need. So going back to actually answering the question, we can write down Bayes' rule. So what is the probability of having cancer given that we have a positive test result? Well that's the probability of getting a positive test result given that we have cancer, multiplied by the probability of having cancer overall. Divided by the probability of a positive test overall. Okay. So we actually know this terms straight out, we're given this one, which is 80%. And we're given this one, the probability of cancer overall is 1%. We don't have this denominator given so we need to figure that out. Well, that's the chance of a positive test in all other occurrences. The chance of a positive test given that we had cancer, and the chance of a positive test given that they don't have cancer. In each case multiplied by the probability of that happening, so you can decompose this. So that's 0.8 times the 1% probability of actually having cancer plus 9.6% times 99%, which gives you this number of 10.3%, so that's the overall probability of getting a positive test result. So, now you plug this in and you end up with a number 0.078% for our answer. If we have a positive test result, the chance of actually having cancer is 7.8%. So this is lower than you might come up with if you don't think about it carefully. Right, you might think that boy I got a positive test result or something. There's this 80% number floating around, there's probably a 70-80 percent chance that I have cancer. But because of the very low percentage of having cancer in the prior probabilities, the actual number is still pretty low. So, it's easy to make mistakes with this stuff. Now, this was a remarkably simple case. First of all, we're given all this information. Second of all, there's only two possibilities, they're these sort of binary variables. So, let's think about something a little more complicated. [MUSIC]