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# Chapter 2 Lecture & Textbook Notes MIE 273

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This 2 page Class Notes was uploaded by Kirin Anandsongvit on Sunday October 4, 2015. The Class Notes belongs to MIE 273 at University of Massachusetts taught by Mi Hyun Park in Fall 2015. Since its upload, it has received 14 views. For similar materials see Probability and Statistics in Mechanical Engineering at University of Massachusetts.

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Date Created: 10/04/15

21 Sample Spaces and Events Keywords Experiment Any activity or process whose outcome is subject to uncertainty Sample Space The sample space of an experiment denoted by 8 is the set of all possible outcomes of that experiment Elements of a sample space are usually enclosed by braces For example When tossing a coin the sample space would consists of either heads or tails The sample space would thus be 8 HT Event An event is any collection subset of outcomes contained in the sample space 8 An event is simply if it consists of exactly one outcome and compound if it consists of more than one outcome The complement of an event A denoted by A is the set of all outcomes in 8 that are not contained in A The union of two events A and B denoted by A U B and read A or B is the event consisting of all outcomes that are either in A or in B or in both events so that the union includes outcomes for which both A and B occur as well as outcomes for which exactly one occursthat is all outcomes in at least one of the events The intersection of two events A and B denoted by A H B and read A and B is the event consisting of all outcomes that are in both A and B For example Let A 01 234 B 3456 and C 135 then A 56 AU B O123456 5 AUC 012345 An B 34 Anc 13 An C O2456 Null Event Let denote the null event the event consisting of no outcomes whatsoever When A H B A and B are said to be mutually exclusive or disjoint events The operations of unions and intersections can be extended to more than two events For any three events A B and C the event AU BU C is the set of outcomes contained in at least one of the three events whereas A B C is the set of outcomes contained in all three events A pictorial representation of events and manipulations is obtained by using Venn Diagrams 22 Axioms Interpretations and Properties of Probability Given an experiment and sample space 8 the objective of probability is to assign to each event A a number PA called the probability of the event A which will give a precise measure of the chance that A will occur Axiom 1 For an event A PA 20 The intuitive notion that the chances ofA occurring should be nonnegative Axiom 2 PS 1 The maximum possible probability of 1 is assigned to 8 Axiom 3 If A1 1213 is an infinite collection of disjoint events then PAl UAZUA3U z PAl i1 If we wish the probability that at least one of a number of events will occur and no two of the events will occur simultaneously then the chance of at least one occurring is the sum of the chances of the individual elements

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