Robert P. Dobrow - Probability

In probability word problems, descriptive phrases are typically used rather than set notation. See Table 1.2for some equivalences.

A Venn diagram is a useful tool for working with events and subsets. A rectangular box denotes the sample space , and circles are used to denote events. See Figure 1.1for examples of Venn diagrams for the most common combined events obtained from two events and .

One of the most basic, and important, properties of a probability function is the simple addition rule for mutually exclusive events. We say that two events are mutually exclusive , or disjoint , if they have no outcomes in common. That is, and are mutually exclusive if , the empty set.

TABLE 1.2 . Events and sets.

Description	Set notation
Either or both occur
Not
implies ; is a subset of
Neither nor
At least one of the two events occurs
At most one of the two events occurs

FIGURE 1.1 : Venn diagrams.

If and are mutually exclusive events, then

The addition rule is a consequence of the third defining property of a probability function. We have that

where the third equality follows because the events are disjoint, so no outcome will be counted twice. The addition rule for mutually exclusive events extends to more than two events.

Suppose is a sequence of pairwise mutually exclusive events. That is, and are mutually exclusive for all . Then

Next, we highlight other key properties that are consequences of the defining properties of a probability function and the addition rule for disjoint events.

1 If implies , that is, if , then

2

3 For all events and , (1.3)

Each property is derived next.

1 As , write as the disjoint union of and . By the addition rule for disjoint events,because probabilities are nonnegative.

2 The sample space can be written as the disjoint union of any event and its complement . Thus,Rearranging gives the result.

3 Write as the disjoint union of and . Also write as the disjoint union of and . Then and thus,Observe that the addition rule for mutually exclusive events follows from Property 3 because if and are disjoint, then .