Explain Joint probability

It is simple to work on mutually exclusive cases but most of the actual problems belong to non-mutually exclusive events. We can predict the event outcome by using the joint appearance. For example, if emails messages present the word like lottery, which is very highly likely of being spam rather than ham. The following Venn diagram indicates the joint probability of spam with lottery. However, lottery circle is not contained completely within the spam circle. This implies that not every email with the word lottery is spam and not all spam messages contain the word lottery.

In the following diagram, we have expanded Venn diagram representation:

From the above representation,10 percent of all the emails are spam and 4 percent of emails have the word lottery and our task is to quantify the degree of overlap between these two proportions. In other words, we need to identify the joint probability of spam and lottery occurring which can also be written as p(spam lottery. The respective value can be found as p(spam ∩ lottery) = p(spam) * p(lottery) = 0.1 * 0.04 = 0.004, which is 0.4 percent of all messages are spam containing the word Lottery. In general, for independent events are calculated as P(A∩ B) = P(A) * P(B).