Continuing our discussion of probability, the next topic I want to look at is Bayes Theorem. And Bayes Theorem states that the probability that an event B will occur, given that some other event A has already occurred, when A and B are dependent or are given by this equation here. This is most easy to illustrate, this is not a simple concept, but let's do this by means of this example. Let's suppose that we have a rare disease which afflicts one in 1,000 adults. And let's suppose that if the person does have the disease, there is a test which will show positive 99% of the time. But on the other hand, if the person does not have the disease the result will show positive 2% of the time. That's called a false positive. For a randomly tested person, the test's positive. The probability that that person has the disease is which of these four alternatives? So, this is a problem that we'll utilize Bayes Theorem that we've already given. The probability that the event B will occur, given that the even A has already occurred when they're both dependent, is given by this equation which we've already had. So, to use that, let's suppose that A1 is the event that the individual does have the disease, and A2 is that the individual does not have the disease, and B is a positive test result. So, then we have the probability of A1, in other words, the probability that the person does have the disease is 1 in 1000 In other words, .001. On the other hand, the probability of A2, in other words the probability that the individual does not have the disease, is one minus that, or point 0.999 and the probability of B occurring when A1 has occurred is 99%, this one here. In other words, the probability of B when A1 has occurred is .99, probability of a positive result when the person does have the disease. Conversely, the probability of B occurring when A2 occurs is .02, 2% of the time. In other words, the probability of a positive result when the person does not have the disease is .02. So now we can substitute these values into our basic equation for Bayes Theorem which then looks like this. Probability of A1 is .001 P B A2, is 0.99, etc, .99, substituting in the numbers, the answer is 0.0472, and the answer is C. And a final note that you also see this notation sometimes used for the Bayes Theorem probability. So the result here is completely counter intuitive that the person has the disease and tests positive. But in fact, the probability of them having it is only about 5%. A completely counter intuitive result which you would have intuitively guessed. And the reason that that occurs in this case is that the overall probability of it occurring here, the probability of actually being positive, is only one-tenth of a percent. If that probability had been much higher, then the probability of the positive test result would have been much closer. So this concludes my discussion of Bayes Theorem.
