13C - The addition rule of probability
Learning intentions:
In this section we will examine:
- Set notation (a review from 1A).
- 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.
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:
- ...
Success criteria:
You will be successful if you can:
- Identify elements in the intersection or union of two events.
- State the element in the complementary set of an event.
- Use set theory to solve probability problems.
- Use the addition rule for combining probabilities to solve complex problems.