Chain rule of probability

In the last article, we discussed the concept of conditional probability and we know that the formula for computing the conditional probability is as follows:

We can re-arrange the terms in this formula

Similarly, we have the formula for P(B | A)

Let’s take an example to understand the above formula, say the sample space is represented by omega and represents the population of a country and set A is the set of people who are infected by COVID-19, and set B is the set of people for whom the test results have come out to be positive

In such scenarios, 4 events are of interest to us:

The first one is A intersection B which represents the people who have the COVID-19 symptoms and their test results have come out to be positive and that lies in the intersection region as highlighted in the below image:

Then we have the set of the population such that the people are not infected but the test results have come out to be positive say because of some error in the test, so that part of B which overlaps with A complement and this region is highlighted in the below image

Then the next one is the set of people who are actually COVID-19 infected but their test results have not come out as positive because of test error and the highlighted region in the below image represents such population: