* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Probability
Survey
Document related concepts
History of randomness wikipedia , lookup
Indeterminism wikipedia , lookup
Random variable wikipedia , lookup
Probabilistic context-free grammar wikipedia , lookup
Dempster–Shafer theory wikipedia , lookup
Infinite monkey theorem wikipedia , lookup
Probability box wikipedia , lookup
Birthday problem wikipedia , lookup
Boy or Girl paradox wikipedia , lookup
Inductive probability wikipedia , lookup
Ars Conjectandi wikipedia , lookup
Transcript
3-1 of 18 Chapter Three Probability McGraw-Hill/Irwin Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved. 3-2 of 18 Probability 3.1 3.2 3.3 3.4 The Concept of Probability Sample Spaces and Events Some Elementary Probability Rules Conditional Probability and Independence 3-3 of 18 3.1 Probability Concepts An experiment is any process of observation with an uncertain outcome. The possible outcomes for an experiment are called the experimental outcomes. Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out 3-4 of 18 Probability If E is an experimental outcome, then P(E) denotes the probability that E will occur and Conditions 0 P( E ) 1 If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 The probabilities of all the experimental outcomes must sum to 1. Interpretation: long-run relative frequency or subjective 3-5 of 18 3.2 The Sample Space The sample space of an experiment is the set of all experimental outcomes. Example 3.2: Genders of Two Children 3-6 of 18 Computing Probabilities of Events An event is a set (or collection) of experimental outcomes. The probability of an event is the sum of the probabilities of the experimental outcomes that belong to the event. 3-7 of 18 Example: Computing Probabilities Example 3.4: Genders of Two Children Events P(one boy and one girl) = P(BG) + P(GB) = ¼ + ¼ = ½ P(at least one girl) = P(BG) + P(GB) + P(GG) = ¼ + ¼ + ¼ = ¾ Note: Experimental Outcomes: BB, BG, GB, GG All outcomes equally likely: P(BB) = … = P(GG) = ¼ 3-8 of 18 Probabilities: Equally Likely Outcomes If the sample space outcomes (or experimental outcomes) are all equally likely, then the probability that an event will occur is equal to the ratio the number of sample space outcomes that correspond to the event The total number of sample space outcomes 3-9 of 18 Example: AccuRatings Case Of 5528 residents sampled, 445 prefer KPWR. Estimated Share: P(KPWR) = 445/5528 = 0.0805 Assuming 8,300,000 Los Angeles residents aged 12 or older: Listeners = Population x Share = 8,300,000 x 0.08 = 668,100 3-10 of 18 3.3 Event Relations The complement A of an event A is the set of all sample space outcomes not in A. Further, P(A) = 1 - P(A) Union of A and B, A B Elementary events that belong to either A or B (or both.) Intersection of A and B, A B Elementary events that belong to both A and B. 3-11 of 18 The Addition Rule for Unions The probability that A or B (the union of A and B) will occur is P(A B) = P(A) + P(B) - P(A B) A and B are mutually exclusive if they have no sample space outcomes in common, or equivalently if P(A B) = 0 3-12 of 18 3.4 Conditional Probability The probability of an event A, given that the event B has occurred is called the “conditional probability of A given B” and is denoted as . Further, P(A | B) P(A B) P(A|B) = P(B) 3-13 of 18 Independence of Events Two events A and B are said to be independent if and only if: P(A|B) = P(A) or, equivalently, P(B|A) = P(B) 3-14 of 18 The Multiplication Rule for Intersections The probability that A and B (the intersection of A and B) will occur is P(A B) = P(A) P(B | A) = P(B) P(A | B) If A and B are independent, then the probability that A and B (the intersection of A and B) will occur is P(A B) = P(A) P(B) P(B) P(A) 3-15 of 18 Contingency Tables P(R1 ) P(R1 C1 ) R1 R2 Total P(R 2 C2 ) C1 .4 .1 .5 C2 .2 .3 .5 Total .6 .4 1.00 P(C 2 ) 3-16 of 18 Example: AccuRatings Case Example 3.16: Estimating Radio Station Share by Daypart 5528 L.A. residents sampled. 2827 of residents sampled listen during some portion of the 6-10 a.m. daypart. Of those, 201 prefer KIIS. KIIS Share for 6-10 a.m. daypart: P(KIIS|6-10 a.m.) = P(KIIS 6-10 a.m.) / P(6-10 a.m.) = (201/5528) (2827/5528) = 201/2827 = 0.0711 3-17 of 18 Probability Summary: 3.1 3.2 3.3 3.4 The Concept of Probability Sample Spaces and Events Some Elementary Probability Rules Conditional Probability and Independence 3-18 of 18