Few relevant information which is important

Consider Bi are mutually exclusive and exhaustive events, such that , which is the sure event, i.e., technically the whole of the sample space and also assume ,  is true. Now suppose  and  are events, such that we write  as . Then what may be of interest to us is to find

Now due to the fact that  is true, we can write the following, i.e., , so with conditional probability  now becomes

Pictorially we can denote this as following as shown in Figure 1.17.

Figure 1.17: Illustration of the concept of conditional probability and joint distribution

Hence we can write

as

Hence: .