Probability. Robert P. Dobrow
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Название: Probability

Автор: Robert P. Dobrow

Издательство: John Wiley & Sons Limited

Жанр: Математика

Серия:

isbn: 9781119692416

isbn:

СКАЧАТЬ href="#fb3_img_img_dfd2059f-0152-59a4-9078-69c4fc647715.png" alt="upper A upper B Superscript c"/> Neither upper A nor upper B upper A Superscript c Baseline upper B Superscript c At least one of the two events occurs upper A union upper B At most one of the two events occurs left-parenthesis upper A upper B right-parenthesis Superscript c Baseline equals upper A Superscript c Baseline union upper B Superscript c Schematic illustration of the different types of Venn diagrams.

      ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS

      If upper A and upper B are mutually exclusive events, then

upper P left-parenthesis upper A or upper B right-parenthesis equals upper P left-parenthesis upper A union upper B right-parenthesis equals upper P left-parenthesis upper A right-parenthesis plus upper P left-parenthesis upper B right-parenthesis period

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

StartLayout 1st Row 1st Column upper P left-parenthesis upper A or upper B right-parenthesis 2nd Column equals upper P left-parenthesis upper A union upper B right-parenthesis equals sigma-summation Underscript omega element-of upper A union upper B Endscripts upper P left-parenthesis omega right-parenthesis 2nd Row 1st Column Blank 2nd Column equals sigma-summation Underscript omega element-of upper A Endscripts upper P left-parenthesis omega right-parenthesis plus sigma-summation Underscript omega element-of upper B Endscripts upper P left-parenthesis omega right-parenthesis 3rd Row 1st Column Blank 2nd Column equals upper P left-parenthesis upper A right-parenthesis plus upper P left-parenthesis upper B right-parenthesis comma EndLayout

      EXTENSION OF ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS

      Suppose upper A 1 comma upper A 2 comma ellipsis is a sequence of pairwise mutually exclusive events. That is, upper A Subscript i and upper A Subscript j are mutually exclusive for all i not-equals j. Then

upper P left-parenthesis at least one of the upper A Subscript i Baseline prime s occurs right-parenthesis equals upper P left-parenthesis union Underscript i equals 1 Overscript infinity Endscripts upper A Subscript i Baseline right-parenthesis equals sigma-summation Underscript i equals 1 Overscript infinity Endscripts upper P left-parenthesis upper A Subscript i Baseline right-parenthesis period

      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?

      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,

 image

Schematic illustration of a Venn diagram.