1.5.3 Axiom of Additivity

This axiom states that the probability of the union of mutually exclusive events is the sum of the probabilities of the individual events. Two events: ,  are considered to be mutually exclusive if they have no outcome in common.

If , then:

        

Recall that events are sets (Axiom of normalization, Section 1.4.2). Also, If two sets  and  have no elements in common, then  and  are said to be disjoint and their intersection is an empty set, denoted by    =  (Intersection of sets, Section 1.3.7.2). So in the language of sets, we can say that the probability of the union of disjoint sets is the sum of probabilities of the individual sets.

Example 1.26. Though we don’t need proof of an axiom, this example strengthens our understanding of the axiom. Consider a school that provides music and sports clubs as extracurricular activities. In a class consisting of 10 children, each child has the option to join just one of these two clubs. The children made the following enrollment choices:

• 5 children have joined the sports club.
• 4 children have joined the music club.

Now, let's determine the probability that a randomly selected child from this class is a member of either the sports club or the music club.

Since a child cannot simultaneously join both clubs, the probability of a child being in both clubs is 0. So, we are just interested in finding the probability of two mutually exclusive events: Event , representing "a child joining the sports club" and Event , signifying "a child joining the music club". Fig 1.14 illustrates the Venn diagram depicting these events.

Due to the absence of children being members of both clubs, there is no overlap between the circles representing these two events (Fig 1.14).

The probability of these two mutually exclusive events can be calculated as:

        

        

        

        

Which is clearly the sum of their individual probabilities:

                                                   

Thus, we have:

        

Therefore, the probability that a randomly selected child is a member of either the sports or music club is 90%. In other words, 9 out of 10 children have joined either of the two clubs, while 1 has not joined any club.

This axiom has a generic form and extends to more than two events. If  are mutually exclusive events (in other words, the events don't have any outcome in common), then:

        

We know that events are subsets, so the probability of the union of disjoint sets is the sum of probabilities of the individual sets.

The axioms lead to some important consequences which we discuss in the next section.