Though mutually exclusive cases are simple to work upon, most of the actual problems do fall under the category of non-mutually exclusive events. By using the joint appearance, we can predict the event outcome. 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, if you notice in detail, lottery circle is not contained completely within the spam circle. This implies that not all spam messages contain the word lottery and not every email with the word lottery is spam.

In the following diagram, we have expanded the spam and ham category in addition to the lottery word in Venn diagram representation:

We have seen that 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 both p(spam) and p(lottery) occurring, which can be written as p(spam ∩ lottery). In case if both the events are totally unrelated, they are called independent events and their respective value is 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 P(A∩ B) = P(A) * P(B).