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FIGURE 1.1: Venn diagrams.
ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS
If
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
EXTENSION OF ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS
Suppose
Next, we highlight other key properties that are consequences of the defining properties of a probability function and the addition rule for disjoint events.
PROPERTIES OF PROBABILITIES
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 .
Example 1.8 In a city, suppose 75% of the population have brown hair, 40% have brown eyes, and 25% have both brown hair and brown eyes. A person is chosen at random from the city. What is the probability that they
1 Have brown eyes or brown hair?
2 Have neither brown eyes nor brown hair?
To gain intuition, draw a Venn diagram, as in Figure 1.2. Let
1 The probability of having brown eyes or brown hair isNotice that and are not mutually exclusive. If we made a mistake and used the simple addition rule , we would mistakenly get
2 The complement of having neither brown eyes nor brown hair is having brown eyes or brown hair. Thus,
FIGURE 1.2: Venn diagram.
1.5 EQUALLY LIKELY OUTCOMES
The simplest probability model for a finite sample space is that all outcomes are equally likely. If
