b. The intersection of two events A and B, denoted by

Example 2.11 PROPOSITION For any event A, . . If is an infinite collection of disjoint events, then P A 1 ´ A 2 ´ A 3 ´ c 5 g ` i5 1 P A i A 1 , A 2 , A 3 , c P S 5 1 P A 0 You might wonder why the third axiom contains no reference to a finite collection of disjoint events. It is because the corresponding property for a finite collection can be derived from our three axioms. We want our axiom list to be as short as possible and not contain any property that can be derived from others on the list. Axiom 1 reflects the intuitive notion that the chance of A occurring should be non- negative. The sample space is by definition the event that must occur when the exper- iment is performed contains all possible outcomes, so Axiom 2 says that the maximum possible probability of 1 is assigned to . The third axiom formalizes the idea that if we wish the probability that at least one of a number of events will occur and no two of the events can occur simultaneously, then the chance of at least one occurring is the sum of the chances of the individual events. S S where is the null event the event containing no outcomes what- soever. This in turn implies that the property contained in Axiom 3 is valid for a finite collection of disjoint events. [ P [ 5 0 Proof First consider the infinite collection . Since , the events in this collection are disjoint and . The third axiom then gives This can happen only if . Now suppose that are disjoint events, and append to these the infi- nite collection . Again invoking the third axiom, as desired. ■ P a ´ k i 5 1 A i b 5 Pa ´ ` i 5 1 A i b 5 g ` i5 1 P A i 5 g k i5 1 P A i A k1 1 5 [ , A k1 2 5 [ , A k1 3 5 [ , c A 1 , A 2 , c , A k P [ 5 0 P [ 5 gP[ ´ A i 5 [ [ ¨ [ 5 [ A 1 5 [ , A 2 5 [ , A 3 5 [ , c assignments will be consistent with our intuitive notions of probability, all assign- ments should satisfy the following axioms basic properties of probability. AXIOM 1 AXIOM 2 AXIOM 3 Consider tossing a thumbtack in the air. When it comes to rest on the ground, either its point will be up the outcome U or down the outcome D. The sample space for this event is therefore . The axioms specify , so the probability assignment will be completed by determining PU and PD. Since U and D are dis- joint and their union is , the foregoing proposition implies that 1 5 PS 5 PU 1 PD S P S 5 1 S 5 5U, D6 Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook andor eChapters. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.