Probability Notes Math 309 August 20.

Slides:



Advertisements
Similar presentations
Week 21 Basic Set Theory A set is a collection of elements. Use capital letters, A, B, C to denotes sets and small letters a 1, a 2, … to denote the elements.
Advertisements

Chapter 4 Probability and Probability Distributions
Introduction to Probability
Learning Goal 13: Probability Use the basic laws of probability by finding the probabilities of mutually exclusive events. Find the probabilities of dependent.
Class notes for ISE 201 San Jose State University
© Buddy Freeman, 2015Probability. Segment 2 Outline  Basic Probability  Probability Distributions.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 4-2 Basic Concepts of Probability.
Copyright © 2011 Pearson Education, Inc. Probability Chapter 7.
Section 5.2 The Addition Rule and Complements
The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane Probability and Stochastic Processes Yates and Goodman Chapter 1 Summary.
The Erik Jonsson School of Engineering and Computer Science Chapter 1 pp William J. Pervin The University of Texas at Dallas Richardson, Texas
5.1 Basic Probability Ideas
5- 1 Chapter Five McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.
10/1/20151 Math a Sample Space, Events, and Probabilities of Events.
Chapter 8 Probability Section R Review. 2 Barnett/Ziegler/Byleen Finite Mathematics 12e Review for Chapter 8 Important Terms, Symbols, Concepts  8.1.
Chapter 1 Probability Spaces 主講人 : 虞台文. Content Sample Spaces and Events Event Operations Probability Spaces Conditional Probabilities Independence of.
Probability Notes Math 309. Sample spaces, events, axioms Math 309 Chapter 1.
AP Statistics Chapter 6 Notes. Probability Terms Random: Individual outcomes are uncertain, but there is a predictable distribution of outcomes in the.
CPSC 531: Probability Review1 CPSC 531:Probability & Statistics: Review Instructor: Anirban Mahanti Office: ICT Class.
3. Counting Permutations Combinations Pigeonhole principle Elements of Probability Recurrence Relations.
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.
Chapter 1 Fundamentals of Applied Probability by Al Drake.
Random Experiment Random Variable: Continuous, Discrete Sample Space: S Event: A, B, E Null Event Complement of an Event A’ Union of Events (either, or)
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Review of Exam I Sections Jiaping Wang Department of Mathematical Science 02/18/2013, Monday.
CS201: Data Structures and Discrete Mathematics I
PROBABILITY, PROBABILITY RULES, AND CONDITIONAL PROBABILITY
Probability You’ll probably like it!. Probability Definitions Probability assignment Complement, union, intersection of events Conditional probability.
12/7/20151 Math b Conditional Probability, Independency, Bayes Theorem.
YMS Chapter 6 Probability: Foundations for Inference 6.1 – The Idea of Probability.
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.
Probability. Rules  0 ≤ P(A) ≤ 1 for any event A.  P(S) = 1  Complement: P(A c ) = 1 – P(A)  Addition: If A and B are disjoint events, P(A or B) =
Probability: Terminology  Sample Space  Set of all possible outcomes of a random experiment.  Random Experiment  Any activity resulting in uncertain.
Sixth lecture Concepts of Probabilities. Random Experiment Can be repeated (theoretically) an infinite number of times Has a well-defined set of possible.
Chapter 4 Probability, Randomness, and Uncertainty.
+ Chapter 5 Overview 5.1 Introducing Probability 5.2 Combining Events 5.3 Conditional Probability 5.4 Counting Methods 1.
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.
Probability. Randomness When we produce data by randomized procedures, the laws of probability answer the question, “What would happen if we did this.
Chapter 2: Probability. Section 2.1: Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance.
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 4 Probability.
Basic Probability. Introduction Our formal study of probability will base on Set theory Axiomatic approach (base for all our further studies of probability)
6.2 – Probability Models It is often important and necessary to provide a mathematical description or model for randomness.
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Review of Final Part I Sections Jiaping Wang Department of Mathematics 02/29/2013, Monday.
Key Concepts of the Probability Unit Simulation Probability rules Counting and tree diagrams Intersection (“and”): the multiplication rule, and independent.
Probability and Statistics for Computer Scientists Second Edition, By: Michael Baron Chapter 2: Probability CIS Computational Probability and Statistics.
Probability and Statistics for Computer Scientists Second Edition, By: Michael Baron Chapter 2: Probability CIS Computational Probability and.
Lecture Slides Elementary Statistics Twelfth Edition
Math a - Sample Space - Events - Definition of Probabilities
Chapter 3: Probability Topics
Chapter 3 Probability.
Chapter 4 Probability.
Chapter 4 Created by Bethany Stubbe and Stephan Kogitz.
What is Probability? Quantification of uncertainty.
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman
Set Operations Section 2.2.
Probability Models Section 6.2.
Chapter 4 – Probability Concepts
Probability and Statistics for Computer Scientists Second Edition, By: Michael Baron Chapter 2: Probability CIS Computational Probability and.
Chapter 2 Notes Math 309 Probability.
Elementary Statistics 8th Edition
6. Multistage events and application of probability
St. Edward’s University
Probability Notes Math 309.
Mrs.Volynskaya Alg.2 Ch.1.6 PROBABILITY
QUANTITATIVE METHODS 1 SAMIR K. SRIVASTAVA.
General Probability Rules
Probability Rules Rule 1.
Chapter 3 & 4 Notes.
An Introduction to….
Probability Notes Math 309.
Chapter 1 Probability Spaces
Presentation transcript:

Probability Notes Math 309 August 20

Some Definitions Experiment - means of making an observation Sample Space (S) - set of all outcomes of an experiment listed in a mutually exclusive and exhaustive manner Event - subset of a sample space Simple Event - an event which can only happen in one way; (or can be thought of as a sample point - a one element subset of S)

Since events are sets, we need to understand the basic set operations Intersection - everything in A and B Union - everything in A or B or both Complement - everything not in A also denoted by a bar or a prime mark

You should be able to sketch Venn diagrams to describe the intersections, unions, & complements of sets. Note that these set operations obey the commutative, associative, and distributive laws

DeMorgan’s Laws Convince yourself that these are reasonable with Venn diagrams!

Another definition - A and B are mutually exclusive iff A  B = 

Axioms of Probability (these are FACT, no proof needed!) Let E represent an event, S the sample space, Axiom 1: Axiom 2: Axiom 3: For pairwise mutually exclusive events, the probability of their union is the sum of their respective probabilities, i.e.

Sample Spaces with Equally Likely Outcomes In an experiment where all simple events (sample points) are equally likely, one can find the probability of an event by counting two sets.

Complements Unions Intersections

Theorems (You should be able to prove these using the axioms and definitions.) Let A and B be any two events. . Thm 7.1 If , then

Unions get complicated if events are not mutually exclusive! P(A  B  C) = P(A) + P(B) + P(C) - P(A  B) - P(A  C) - P(B  C) + P(A  B  C) B

However, recall For mutually exclusive events the probability of their union is just the sum of their probabilities.

It is sometimes helpful to get mutually exclusive events by intersecting an event with another event and its complement. For example, so that Another helpful observation is that results in mutually exclusive events is:

Intersections & the multiplication rule Apply the multiplication rule to probabilities so that: P(A  B) = P(A)*P(B|A) = P(B)*P(A|B) P(B|A) is read, “the probability of B given A” Tree diagrams may be helpful in visualizing this.

Intersections and  intersection  multiply In general intersections get more complicated when there are more events, e.g. P(ABCD) = P(A)* P(B|A)*P(C|AB)*P(D|A BC) We’ll see which type events become easy for intersections in a later section.

Combinatorial Methods Math 309 August 22

Combinatorics Basic Principle of Counting Permutations Combinations (a.k.a. Multiplication Principle) Permutations Permutations with indistinguishable objects Combinations

Basic Counting Principle If a choice consists of 2 steps where the first m outcomes and the second has n outcomes, then there are m*n outcomes for the whole choice. The principle can be generalized for r steps. The number of outcomes of a choice with r steps is the product of the number of outcomes of each step.

Permutations # of arrangements of one set, order matters application of the basic counting principle where we return to the same set for the next selection P(n,r) = n!/(n-r)!

Combinations the number of selections, order doesn’t matter C(n,r) = n!/[(n-r)!r!] the number of arrangements can be counted by selecting the objects and then ordering them i.e. P(n,r) = C(n,r)*r!

Observations about Combinations C(n, r) = C(n, n-r) C(n, n) = C(n, 0) = 1 C(n, 1) = n = C(n, n-1) C(n, 2) = n(n-1)/2

Permutations with Indistinguishable Objects Order the objects as if they were distinguishable Then “divide out” those arrangements that look identical.

Combinations with Repetition Select r objects from n objects when where each item can be selected more than once. Add n-1 dividers to the r objects to be selected. In the r+n-1 “slots” select the location of the r items, C(r+n-1,r). The blank spaces will denote division of two types of objects.

Combining Counting Techniques If we are careful with language, when we say “AND”, we multiply “AND”  multiplication  intersection when we say “OR”, we add “OR”  addition  union

Conditional Probability & Intersections Math 309 August 27

Conditional Probability P(A|B) P(A|B) is read, “the probability of A given B” B is known to occur.

Intersections and  intersection  multiply In general intersections get more complicated when there are more events, e.g. P(ABCD) = P(A)* P(B|A)*P(C|AB)*P(D|A BC)

Independent Events A and B are independent if any of the following are true: P(AB) = P(A)*P(B) P(A|B) = P(A) P(B|A) = P(B) You need to check probabilities to determine if events are independent. If A, B, C, & D are pairwise independent, P (AB C D) = P(A)*P(B)*P(C)*P(D)

Conditional Probability P(A|B) Formula P(A|B) = P(A  B) / P(B), if P(B) > 0 (Note that this is an algebraic manipulation of the formula for the probability of the intersection of 2 events.) i.e. the conditional probability is the probability that both occur divided by what is given occurs