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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.

      ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS

      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.

      EXTENSION OF ADDITION RULE FOR MUTUALLY EXCLUSIVE 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.

      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 be the event of having brown hair; let Скачать книгу