Introduction

Probability uses the mathematical ideas of sets, as we have seen in the definition of both the sample space of an experiment and in the definition of an event. In order to perform basic probability calculations, we need to review the ideas from set theory related to the set operations of union, intersection, and complement. 

Union

The union of two or more sets is the set that contains all the elements of each of the sets; an element is in the union if it belongs to at least one of the sets. The symbol for union is $\cup$, and is associated with the word "or", because $A \cup B$  is the set of all elements that are in $A$ or $B$ (or both.) To find the union of two sets, list the elements that are in either (or both) sets. In terms of a Venn Diagram, the union of sets $A$ and $B$ can be shown as two completely shaded interlocking circles.

Union of Two Sets

The shaded Venn Diagram shows the union of set $A$ (the circle on left) with set $B$ (the circle on the right). It can be written shorthand as $A \cup B$.

In symbols, since the union of $A$ and $B$ contains all the points that are in $A$ or $B$ or both, the definition of the union is:

$\displaystyle A \cup B = \{x: x\in A \ \text{or} \ x\in B \}$

For example, if $A = \{1, 3, 5, 7\}$ and $B = \{1, 2, 4, 6\}$ , then $A \cup B = \{1, 2, 3, 4, 5, 6, 7\}$. Notice that the element $1$ is not listed twice in the union, even though it appears in both sets $A$ and $B$. This leads us to the general addition rule for the union of two events: 

$\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Where $P(A\cap B)$ is the intersection of the two sets. We must subtract this out to avoid double counting of the inclusion of an element.

If sets $A$ and $B$ are disjoint, however, the event $A \cap B$ has no outcomes in it, and is an empty set denoted as $\emptyset$, which has a probability of zero. So, the above rule can be shortened for disjoint sets only: 

$\displaystyle P(A \cup B) = P(A)+P(B)$

This can even be extended to more sets if they are all disjoint: 

$\displaystyle P(A \cup B \cup C) = P(A) + P(B)+ P(C)$

Intersection

The intersection of two or more sets is the set of elements that are common to each of the sets. An element is in the intersection if it belongs to all of the sets. The symbol for intersection is $\cap$, and is associated with the word "and", because $A \cap B$  is the set of elements that are in $A$ and $B$ simultaneously. To find the intersection of two (or more) sets, include only those elements that are listed in both (or all) of the sets. In terms of a Venn Diagram, the intersection of two sets $A$ and $B$ can be shown at the shaded region in the middle of two interlocking circles .

Intersection of Two Sets

Set $A$ is the circle on the left, set $B$ is the circle on the right, and the intersection of $A$ and $B$, or $A \cap B$, is the shaded portion in the middle.

In mathematical notation, the intersection of $A$ and $B$ is written as $A \cap B = \{x: x \in A \ \text{and} \ x \in B\}$. For example, if $A = \{1, 3, 5, 7\}$ and $B = \{1, 2, 4, 6\}$, then $A \cap B = \{1\}$ because $1$ is the only element that appears in both sets $A$ and $B$.

When events are independent, meaning that the outcome of one event doesn't affect the outcome of another event, we can use the multiplication rule for independent events, which states: 

$\displaystyle P(A \cap B)= P(A)P(B)$

For example, let's say we were tossing a coin twice, and we want to know the probability of tossing two heads. Since the first toss doesn't affect the second toss, the events are independent. Say is the event that the first toss is a heads and $B$ is the event that the second toss is a heads, then $P(A \cap B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$.