13C - The addition rule of probability

Learning intentions:

In this section we will examine:

    1. Set notation (a review from 1A).
    2. The addition rule for combining probabilities.

Review of set notation

Null sets

If sets A and B have no elements in common; that is, there are no elements common to both sets then the intersection is an empty or null set (∅). Furthermore, on the Venn diagram would have no overlap between the two circles representing A and B (see below).

  • Mathematically we can denote the disjoint (null set) A and B as A ∩ B = ∅.

Intersection

The set of elements contained in both sets A and B is called the intersection.

    • We denote the intersection between A and B mathematically as: A ∩ B
    • Note: A ∩ B is equivalent to B ∩ A.

The following is a review of set notion from section 1A.

Intersection

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
Union

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:

Union

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:

    1. ...

Success criteria:

You will be successful if you can:

    1. Identify elements in the intersection or union of two events.
    2. State the element in the complementary set of an event.
    3. Use set theory to solve probability problems.
    4. Use the addition rule for combining probabilities to solve complex problems.