Chapter 4 Probability. Probability Defined A probability is a number between 0 and 1 that measures the chance or likelihood that some event or set of.

Slides:



Advertisements
Similar presentations
© 2003 Prentice-Hall, Inc.Chap 4-1 Basic Probability IE 440 PROCESS IMPROVEMENT THROUGH PLANNED EXPERIMENTATION Dr. Xueping Li University of Tennessee.
Advertisements

Chapter 4 Probability and Probability Distributions
©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin A Survey of Probability Concepts Chapter 5.
Chapter 4 Introduction to Probability
Business and Economics 7th Edition
Chapter 4 Probability.
Chapter 4 Basic Probability
Chapter 4: Basic Probability
Chapter 4 Basic Probability
A Survey of Probability Concepts
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 4-2 Basic Concepts of Probability.
Chapter 4 Probability Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin.
A Survey of Probability Concepts
McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. A Survey of Probability Concepts Chapter 5.
5.1 Basic Probability Ideas
Section 4-3 The Addition Rule. COMPOUND EVENT A compound event is any event combining two or more simple events. NOTATION P(A or B) = P(in a single trial,
A Survey of Probability Concepts Chapter 5. 2 GOALS 1. Define probability. 2. Explain the terms experiment, event, outcome, permutations, and combinations.
10/1/20151 Math a Sample Space, Events, and Probabilities of Events.
UNIT 8: PROBABILITY 7 TH GRADE MATH MS. CARQUEVILLE.
CHAPTER 5 Probability: Review of Basic Concepts
Using Probability and Discrete Probability Distributions
Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 4 Probability.
Probability Theory Pertemuan 4 Matakuliah: F Analisis Kuantitatif Tahun: 2009.
©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin A Survey of Probability Concepts Chapter 5.
Probability & Statistics I IE 254 Exam I - Reminder  Reminder: Test 1 - June 21 (see syllabus) Chapters 1, 2, Appendix BI  HW Chapter 1 due Monday at.
1 1 Slide © 2016 Cengage Learning. All Rights Reserved. Probability is a numerical measure of the likelihood Probability is a numerical measure of the.
Chapter 12 Probability. Chapter 12 The probability of an occurrence is written as P(A) and is equal to.
2-1 Sample Spaces and Events Random Experiments Figure 2-1 Continuous iteration between model and physical system.
2-1 Sample Spaces and Events Random Experiments Figure 2-1 Continuous iteration between model and physical system.
Chapter 4 Probability ©. Sample Space sample space.S The possible outcomes of a random experiment are called the basic outcomes, and the set of all basic.
A Survey of Probability Concepts
1 Chapter 4 – Probability An Introduction. 2 Chapter Outline – Part 1  Experiments, Counting Rules, and Assigning Probabilities  Events and Their Probability.
Basic Business Statistics Assoc. Prof. Dr. Mustafa Yüzükırmızı
©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin A Survey of Probability Concepts Chapter 5.
1 1 Slide © 2007 Thomson South-Western. All Rights Reserved Chapter 4 Introduction to Probability Experiments, Counting Rules, and Assigning Probabilities.
Introduction to Probability 1. What is the “chance” that sales will decrease if the price of the product is increase? 2. How likely that the Thai GDP will.
Probability Rules. We start with four basic rules of probability. They are simple, but you must know them. Rule 1: All probabilities are numbers between.
Copyright © 2014 by McGraw-Hill Higher Education. All rights reserved. Essentials of Business Statistics: Communicating with Numbers By Sanjiv Jaggia and.
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 4-1 Chapter 4 Basic Probability Basic Business Statistics 11 th Edition.
BIA 2610 – Statistical Methods
Chapter 4 Probability, Randomness, and Uncertainty.
4-3 Addition Rule This section presents the addition rule as a device for finding probabilities that can be expressed as P(A or B), the probability that.
STATISTICS 6.0 Conditional Probabilities “Conditional Probabilities”
1 Lecture 4 Probability Concepts. 2 Section 4.1 Probability Basics.
1 C.M. Pascual S TATISTICS Chapter 5b Probability Addition Rule.
BUSA Probability. Probability – the bedrock of randomness Definitions Random experiment – observing the close of the NYSE and the Nasdaq Sample.
Copyright ©2004 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4-1 Probability and Counting Rules CHAPTER 4.
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 4 Probability.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by.
Probability and Probability Distributions. Probability Concepts Probability: –We now assume the population parameters are known and calculate the chances.
Lecture Slides Elementary Statistics Twelfth Edition
Introduction To Probability
Math a - Sample Space - Events - Definition of Probabilities
Unit 8 Probability.
A Survey of Probability Concepts
Chapter 3 Probability.
Chapter 4 Probability.
Chapter 4 Basic Probability.
Statistics 300: Introduction to Probability and Statistics
A Survey of Probability Concepts
CHAPTER 4 (Part A) PROBABILITY
A Survey of Probability Concepts
Chapter 4 – Probability Concepts
Chapter 4 Basic Probability.
Elementary Statistics 8th Edition
CHAPTER 4 PROBABILITY Prem Mann, Introductory Statistics, 8/E Copyright © 2013 John Wiley & Sons. All rights reserved.
Chapter 4 Probability 4.2 Basic Concepts of Probability
Probability.
Basic Probability Chapter Goal:
Presentation transcript:

Chapter 4 Probability

Probability Defined A probability is a number between 0 and 1 that measures the chance or likelihood that some event or set of events will occur.

Assigning Basic Probabilities Classical Approach Relative Frequency Approach Subjective Approach

Classical Approach P(A) = where P(A) = probability of event A F = number of outcomes “favorable” to event A T = total number of outcomes possible in the experiment

Relative Frequency Approach P(A) = where N = total number of observations or trials n = number of times that event A occurs

The Language of Probability Simple Probability Conditional Probability Independent Events Joint Probability Mutually Exclusive Events Either/Or Probability

Statistical Independence Two events are said to be statistically independent if the occurrence of one event has no influence on the likelihood of occurrence of the other.

Statistical Independence (4.1) P(A l B) = P(A) and P (B l A) = P(B) “given”

General Multiplication Rule (4.2) P(A  B) = P(A)∙ P(BlA) “and”

Multiplication Rule for (4.3) Independent Events P(A  B) = P(A) ∙P(B)

Mutually Exclusive Events Two events, A and B, are said to be mutually exclusive if the occurrence of one event means that the other event cannot or will not occur.

Mutually Exclusive Events (4.4) P(A  B) = 0

General Addition Rule (4.5) P(A  B) = P(A) + P(B) - P(A  B) “or”

Addition Rule for (4.6) Mutually Exclusive Events P(A  B) = P(A) + P(B)

“Conditional Equals JOINT Over SIMPLE” Rule P(B I A) = (4.7a) P(A I B) = (4.7b)

Complementary Events Rule (4.8) P(A′ ) = 1 – P(A)

Figure 4.1 Venn Diagram for the Internet Shoppers Example A(.8) (Airline Ticket Purchase) B(.6) (Book Purchase) A∩ B (.5) The sample space contains 100% of the possible outcomes in the experiment. 80% of these outcomes are in Circle A; 60% are in Circle B; 50% are in both circles. Sample Space (1.0)

Figure 4.2 Complementary Events The events A and A’ are said to be complementary since one or the other (but never both) must occur. For such events, P(A’ ) = 1 - P(A). Sample Space (1.0) A.8 A’ (.2) (everything in the sample space outside A)

Figure 4.3 Mutually Exclusive Events Mutually exclusive events appear as non-overlapping circles in a Venn diagram. Sample Space (1.0) BA

Figure 4.4 Probability Tree for the Project Example B is not under budget A A' A is under budget B is under budget B B' A is not under budget B is under budget B is not under budget B PROJECT A PERFORMANCE PROJECT B PERFORMANCE B' STAGE 1STAGE 2

Figure 4.5 Showing Probabilities on the Tree B is not under budget A (.25) A‘(.75) A is under budget B is under budget B(.6) B‘(.4) A is not under budget B is under budget B is not under budget B(.2) PROJECT A PERFORMANCE PROJECT B PERFORMANCE B‘(.8) STAGE 1STAGE 2

Figure 4.6 Identifying the Relevant End Nodes On The Tree  B is not under budget A (.25) A‘(.75) A is under budget B is under budget B(.6) B‘(.4) A is not under budget B is under budget B is not under budget B(.2) PROJECT A PERFORMANCE PROJECT B PERFORMANCE B‘(.8) STAGE 1STAGE 2 (2) (1) (3) (4) 

Figure 4.7 Calculating End Node Probabilities B is not under budget A (.25) A‘(.75) A is under budget B is under budget B(.6) B‘(.4) A is not under budget B is under budget B is not under budget B(.2) PROJECT A PERFORMANCE PROJECT B PERFORMANCE B‘(.8) STAGE 1STAGE 2 (2) (1) (3) (4).10.15

Figure 4.8 Probability Tree for the Spare Parts Example Unit is OK A 1 A 2 Adams is the supplier Unit is defective B B' Alder is the supplier Unit is defective Unit is OK B B' (.7) (.3) (.04) (.96) (.07) (.93) SOURCECONDITION

Figure 4.9 Using the Tree to Calculate End-Node Probabilities Unit is OK A 1 A 2 Adams is the supplier Unit is defective B B' Alder is the supplier Unit is defective Unit is OK B B' (.7) (.3) (.04 ) (.96) (.07) (.93) SOURCECONDITION .028 .021

Bayes’ Theorem (4.9) (Two Events) P(A 1 l B) =

General Form of Bayes’ Theorem P ( A i l B ) =

Cross-tabs Table Very ImportantImportantNot Important Under Grad Grad

Joint Probability Table Very Important Not Important Under Grad Grad

Counting Total Outcomes (4.10) in a Multi-Stage Experiment Total Outcomes = m 1 x m 2 x m 3 x…x m k where m i = number of outcomes possible in each stage k = number of stages

Combinations (4.11) n C x = where n C x = number of combinations (subgroups) of n objects selected x at a time n = size of the larger group x = size of the smaller subgroups

Figure 4.10 Your Car and Your Friends Five friends are waiting for a ride in your car, but only four seats are available. How many different arrangements of friends in the car are possible? A B E C D 1 Friends Car

Permutations (4.12) nPx =nPx =