13C - The addition rule of probability

Figure 1 - The intersection of A and B (A ∩ B).

Union

The set of elements contained in A and B; that is, all elements in A and all the elements in B, is called the union. In the case of a union, the elements do not have to be in both A and B

(just in one of them).

• Mathematically we express the union of sets A and B as: A ∪ B

Figure 2 - The union of A and B (A ∪ B).

Complementary events

The complementary set of A is all of the element in not in set A.

• Mathematically we represent the complement of A as A'.
• The union of two complementary sets, A ∪ A', will contain all elements in the system.

Figure 3 - The complement of A (A').

Mutually exclusive

If two sets (or events in probability), A and B have no elements in common then they are considered mutually exclusive.

Figure 3 - Two mutually exclusive events A and B (A ∩ B = ∅).

The addition rule of probability

The addition rule of probability allows us to calculate the probability of the union between two events. Consider the following Venn diagram for events A and B:

Given that the intersection of A and B is not the null set (∅), the number of elements in the union is given by:

The element in the intersection were counted twice, once in n(A) and again in n(B); thus, we need to subtract the intersection. By dividing each term by the total number of elements in the sample space we find the addition rule for combing probabilities:

This rule is especially useful along with Karnaugh maps for solving complex problems.

1A - Exercises:

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