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In this chapter, you will learn to: Explain the terms random experiment, outcome, sample space, permutations, and combinations; define probability; describe the classical, empirical, and subjective approaches to probability; explain and calculate conditional probability and joint probability;...
5 1 A Survey of Concepts Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 2 When you have completed this chapter, you will be able to: Explain the terms random experiment, outcome, sample space, permutations, and combinations Define probability Describe the classical, empirical, and subjective approaches to probability Explain and calculate conditional probability and joint probability. Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 3 Calculate probability using the rules of addition and rules of multiplication Use a tree diagram to organize and compute probabilities Calculate a probability using Bayes’ theorem Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Types of Statistics Types of Statistics Descriptive Descriptive Methods of… collecting organizing presenting and analyzing data Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 4 Inferential Inferential Science of… Science of… making inferences making inferences about a population, about a population, based on based on sample sample information information Emphasis now to be on this! Emphasis now to be on this! Terminology 5 5 Probability …is a measure of the …is a measure of the likelihood that an event in the future will happen! likelihood that an event in the future will happen! It can only assume a value between 0 and 1 A value near zero means the event is not likely happen; near one means it is likely There are three definitions of probability: classical, empirical, and subjective Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology 5 6 Random Experiment …is a process …is a process repetitive in nature the outcome of any trial is uncertain welldefined set of possible outcomes each outcome has a probability associated with it Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology 5 7 …is a particular result of a random experiment . is the collection or set of all the possible outcomes of a random experiment …is the collection of one or more outcomes of an experiment Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Approaches to Assigning Probability 5 8 ubjective SSubjective …probability is based on whatever information is available …probability is based on whatever information is available Objective bjective O Classical lassical PProbability robability C … is based on the is based on the … assumption that the assumption that the outcomes of an experiment outcomes of an experiment are equally likely are equally likely Empirical mpirical P Probability robability E … applies when the number … applies when the number of times the event happens of times the event happens is divided by the number of is divided by the number of observations observations Probability Probability NUMBER of favourable outcomes = Total NUMBER of possible outcomes of an Event an Event of Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Examples Examples S ubjective Probability 5 9 …. refers to the chance of occurrence chance of occurrence …. refers to the assigned to an event assigned to an event by a particular individual by a particular individual It is not computed objectively not computed objectively, , It is i.e., not not from prior knowledge or from actual from prior knowledge or from actual i.e., data… data… …that the Toronto Maple Leafs will win the Stanley Cup next season! …that you will arrive to class on time tomorrow! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. E mpirical Probability 5 10 Students measure the contents of their soft Students measure the contents of their soft drink cans… 10 cans are underfilled, drink cans… 10 cans are underfilled, 32 are filled correctly and 32 are filled correctly and When the contents of the next can is measured, When the contents of the next can is measured, 8 are overfilled 8 are overfilled what is the probability that it is… (a) filled correctly? what is the probability that it is… (a) filled correctly? P(C) = 32 / 50 = 64% …(b) not filled correctly? …(b) not filled correctly? P(~C) = 1 – P(C) = 1 .64 = 36% This is called the Complement of C Complement of C This is called the Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Terminology 5 44 Independent Events Draw three cards with replacement Draw three cards with replacement i.e., draw one card, i.e., draw one card, look at look at it, it, put it back, put it back, and repeat and repeat Each draw is independent of the other Each draw is independent of the other twice more twice more Find the probability of drawing 3 Queens in a row: P(3Q) = 4/52 * 4/52 *4/52 = 0.00046 = = 0.00046 = most unlikely! most unlikely! P(3Q) = 4/52 * 4/52 *4/52 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Independent Events 5 45 Consider 2 events: Consider 2 events: Drawing a RED card from a deck of cards Drawing a RED card from a deck of cards Drawing a HEART from a deck of cards Drawing a HEART from a deck of cards Are these two events considered to be independent? If two events, A and B are independent, then If two events, A and B are independent, then P(A|B) = P(A) P(A|B) = P(A) P(Red) = 26/52 = 1/2 P(Red|Heart) = 13/13 = 1 Therefore these are NOT independent events! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 46 ayes’ heorem Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ayes’ 5 47 heorem …is a method for revising a probability …is a method for revising a probability given additional information! given additional information! Formula Formula P(A1|B) = P(A1 ) P(B|A1 ) P(A1 ) P(B|A1)+ P(A2 ) P(B|A2 ) Example Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 48 ayes’ heorem Duff Cola Company recently received several complaints that their bottles are underfilled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. What is the probability that the under What is the probability that the under filled bottle came from plant A? filled bottle came from plant A? A B % of Total Production % of Underfilled Bottles 55 45 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 49 ayes’ heorem What is the probability that the under What is the probability that the under filled bottle came from plant A? filled bottle came from plant A? A B % of Total Production % of Underfilled Bottles 55 45 List the List the robabilities given PProbabilities given P(plant A) = .55 P(plant A) = .55 Input values into Input values into formula and compute formula and compute P(Underfilled A) = .03 P(Underfilled A) = .03 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. P(plant B) = .45 P(plant B) = .45 P(Underfilled B) = .04 P(Underfilled B) = .04 ayes’ 5 50 heorem What is the probability that the under What is the probability that the under filled bottle came from plant A? filled bottle came from plant A? List the List the robabilities given PProbabilities given P(plant A) = .55 P(plant A) = .55 Input values into Input values into formula and compute formula and compute P(Underfilled/A) = .03 P(Underfilled/A) = .03 P(A1 ) P(B|A1 ) P(A1 |B) = P(A1 )P(B|A1 )+ P(A2 ) P(B|A2 ) 55(.03) = 55(.03) + 45(.04) = .4783 = .4783 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. P(plant B) = .45 P(plant B) = .45 P(Underfilled/B) = .04 P(Underfilled/B) = .04 The likelihood that the The likelihood that the underfilled bottle came underfilled bottle came from Plant A from Plant A has been reduced from has been reduced from 55% to 55% to 47.83% 47.83% 5 51 Counting Rules Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. actorials ! 5 52 … this is just a shorthand notation that is sometimes used to save time! Examples: Examples: 5! … Means 5*4*3*2*1 = 120 5! … Means 5*4*3*2*1 = 120 4! … Means 4*3*2*1 = 24 4! … Means 4*3*2*1 = 24 By definition, 1! =1 and 0! =1 By definition, 1! =1 and 0! =1 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 53 ermutation …is a counting technique …is a counting technique that is used when order order is is that is used when n! important! important! P n r = (n – r)! ombination …is a counting technique …is a counting technique that is used when order is NOT important! order is NOT important! that is used when n! n Cr = r!(n – r)! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 54 ermutation n! n Pr = (n – r)! …How many ways can you arrange nn things, things, …How many ways can you arrange taking rr at a time, when order is important? at a time, when order is important? taking Example: Example: You are assigned the task of choosing 2 2 of your of your 6 6 classmates to classmates to You are assigned the task of choosing serve on a task force. One will act as the serve on a task force. One will act as the Chair of the task force, and the other will be the Chair of the task force, and the other will be the Secretary. In how many Secretary. In how many ways can you make this assignment? ways can you make this assignment? P2 = 6! / (62)! = 6! / 4! = 6*5 = 30 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ombination 5 55 …is a counting technique …is a counting technique that is used when order is order is that is used when NOT important! NOT important! n! n Cr = r(n – r)! Example: Example: You are assigned the task of choosing 2 2 of your of your 6 6 classmates to classmates to You are assigned the task of choosing serve on a task force. Responsibilities are evenly shared. serve on a task force. Responsibilities are evenly shared. In how many ways can you make this assignment? In how many ways can you make this assignment? C2 = 6! / (2!(62)!) = 6! /2!4! = (6*5)/2 = 15 Using… Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Using… i 5 56 Texas Instruments BAII PLUS ombination 30 15 ermutation 6 nCr nPr 2 15 15 30 30 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. Test your learning… … Test your learning 5 57 … … n o n o k ilcick CCl www.mcgrawhill.ca/college/lind Online Learning Centre for quizzes extra content data sets searchable glossary access to Statistics Canada’s EStat data …and much more! Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. 5 58 This completes Chapter 5 Copyright © 2004 by The McGrawHill Companies, Inc. All rights reserved. ... Diamonds Clubs What is the probability of drawing one card that is both a What is the probability of drawing one card that is both a Jack and a King from a deck of wellshuffled cards? Jack and a King from a deck of wellshuffled cards?... 4 Suits (13 cards in each) Scenarios Hearts Spades Diamonds Clubs What is the probability of drawing a King given that you given that you What is the probability of drawing a King have drawn a BLACK card?... What is the probability of drawing one card that is both What is the probability of drawing one card that is both BLACK and a King from a deck of wellshuffled cards? BLACK and a King from a deck of wellshuffled cards? P( Black and King) =2/52