Chapter 7: Counting Principles

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
Chapter Two Probability
Chapter 7 Probability. Definition of Probability What is probability? There seems to be no agreement on the answer. There are two broad schools of thought:
LING 438/538 Computational Linguistics Sandiway Fong Lecture 17: 10/24.
Lecture II.  Using the example from Birenens Chapter 1: Assume we are interested in the game Texas lotto (similar to Florida lotto).  In this game,
Sets, Combinatorics, Probability, and Number Theory Mathematical Structures for Computer Science Chapter 3 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesProbability.
Sets, Combinatorics, Probability, and Number Theory Mathematical Structures for Computer Science Chapter 3 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesProbability.
10/1/20151 Math a Sample Space, Events, and Probabilities of Events.
PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Independence and Bernoulli.
Chapter 8 Probability Section R Review. 2 Barnett/Ziegler/Byleen Finite Mathematics 12e Review for Chapter 8 Important Terms, Symbols, Concepts  8.1.
DISCRETE COMPUTATIONAL STRUCTURES CSE 2353 Summer 2005.
Nor Fashihah Mohd Noor Institut Matematik Kejuruteraan Universiti Malaysia Perlis ІМ ќ INSTITUT MATEMATIK K E J U R U T E R A A N U N I M A P.
Principles of Statistics Chapter 2 Elements of Probability.
Discrete Mathematical Structures (Counting Principles)
Chapter 1:Independent and Dependent Events
Random Variables. A random variable X is a real valued function defined on the sample space, X : S  R. The set { s  S : X ( s )  [ a, b ] is an event}.
DISCRETE COMPUTATIONAL STRUCTURES CS Fall 2005.
3. Counting Permutations Combinations Pigeonhole principle Elements of Probability Recurrence Relations.
Week 11 What is Probability? Quantification of uncertainty. Mathematical model for things that occur randomly. Random – not haphazard, don’t know what.
EQT 272 PROBABILITY AND STATISTICS
Copyright © Cengage Learning. All rights reserved. Elementary Probability Theory 5.
PROBABILITY, PROBABILITY RULES, AND CONDITIONAL PROBABILITY
Basic Principles (continuation) 1. A Quantitative Measure of Information As we already have realized, when a statistical experiment has n eqiuprobable.
Probability You’ll probably like it!. Probability Definitions Probability assignment Complement, union, intersection of events Conditional probability.
Discrete Structures By: Tony Thi By: Tony Thi Aaron Morales Aaron Morales CS 490 CS 490.
Probability: Terminology  Sample Space  Set of all possible outcomes of a random experiment.  Random Experiment  Any activity resulting in uncertain.
Probability Definition : The probability of a given event is an expression of likelihood of occurrence of an event.A probability isa number which ranges.
Introduction  Probability Theory was first used to solve problems in gambling  Blaise Pascal ( ) - laid the foundation for the Theory of Probability.
Probability Basics Section Starter Roll two dice and record the sum shown. Repeat until you have done 20 rolls. Write a list of all the possible.
Sixth lecture Concepts of Probabilities. Random Experiment Can be repeated (theoretically) an infinite number of times Has a well-defined set of possible.
+ Chapter 5 Overview 5.1 Introducing Probability 5.2 Combining Events 5.3 Conditional Probability 5.4 Counting Methods 1.
Lecture 7 Dustin Lueker.  Experiment ◦ Any activity from which an outcome, measurement, or other such result is obtained  Random (or Chance) Experiment.
Probability. 3.1 Events, Sample Spaces, and Probability Sample space - The set of all possible outcomes for an experiment Roll a die Flip a coin Measure.
Probability theory is the branch of mathematics concerned with analysis of random phenomena. (Encyclopedia Britannica) An experiment: is any action, process.
STATISTICS 6.0 Conditional Probabilities “Conditional Probabilities”
1 Probability- Basic Concepts and Approaches Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND.
Basic Probability. Introduction Our formal study of probability will base on Set theory Axiomatic approach (base for all our further studies of probability)
PROBABILITY 1. Basic Terminology 2 Probability 3  Probability is the numerical measure of the likelihood that an event will occur  The probability.
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
Mathematics Department
Virtual University of Pakistan
ICS 253: Discrete Structures I
Chapter Two Probability
Math a - Sample Space - Events - Definition of Probabilities
Chapter 11 Probability.
PROBABILITY AND PROBABILITY RULES
4 Elementary Probability Theory
What is Probability? Quantification of uncertainty.
9. Counting and Probability 1 Summary
Probability The term probability refers to indicate the likelihood that some event will happen. For example, ‘there is high probability that it will rain.
CHAPTER 4 (Part A) PROBABILITY
Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman
4 Elementary Probability Theory
Basic Concepts An experiment is the process by which an observation (or measurement) is obtained. An event is an outcome of an experiment,
STA 291 Spring 2008 Lecture 7 Dustin Lueker.
Probability Models Section 6.2.
CONDITIONAL PROBABILITY
Welcome to the wonderful world of Probability
Chapter 2.3 Counting Sample Points Combination In many problems we are interested in the number of ways of selecting r objects from n without regard to.
Sets and Probabilistic Models
Independence and Counting
Sets and Probabilistic Models
Independence and Counting
Independence and Counting
Sets and Probabilistic Models
A random experiment gives rise to possible outcomes, but any particular outcome is uncertain – “random”. For example, tossing a coin… we know H or T will.
Probability.
Sets, Combinatorics, Probability, and Number Theory
Presentation transcript:

Chapter 7: Counting Principles Discrete Mathematical Structures: Theory and Applications

Learning Objectives Learn the basic counting principles— multiplication and addition Explore the pigeonhole principle Learn about permutations Learn about combinations Discrete Mathematical Structures: Theory and Applications

Learning Objectives Explore generalized permutations and combinations Learn about binomial coefficients and explore the algorithm to compute them Discover the algorithms to generate permutations and combinations Become familiar with discrete probability Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles There are three boxes containing books. The first box contains 15 mathematics books by different authors, the second box contains 12 chemistry books by different authors, and the third box contains 10 computer science books by different authors. A student wants to take a book from one of the three boxes. In how many ways can the student do this? Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Suppose tasks T1, T2, and T3 are as follows: T1 : Choose a mathematics book. T2 : Choose a chemistry book. T3 : Choose a computer science book. Then tasks T1, T2, and T3 can be done in 15, 12, and 10 ways, respectively. All of these tasks are independent of each other. Hence, the number of ways to do one of these tasks is 15 + 12 + 10 = 37. Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Morgan is a lead actor in a new movie. She needs to shoot a scene in the morning in studio A and an afternoon scene in studio C. She looks at the map and finds that there is no direct route from studio A to studio C. Studio B is located between studios A and C. Morgan’s friends Brad and Jennifer are shooting a movie in studio B. There are three roads, say A1, A2, and A3, from studio A to studio B and four roads, say B1, B2, B3, and B4, from studio B to studio C. In how many ways can Morgan go from studio A to studio C and have lunch with Brad and Jennifer at Studio B? Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles There are 3 ways to go from studio A to studio B and 4 ways to go from studio B to studio C. The number of ways to go from studio A to studio C via studio B is 3 * 4 = 12. Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Consider two finite sets, X1 and X2. Then This is called the inclusion-exclusion principle for two finite sets. Consider three finite sets, A, B, and C. Then This is called the inclusion-exclusion principle for three finite sets. Discrete Mathematical Structures: Theory and Applications

Basic Counting Principles Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle. Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Pigeonhole Principle Discrete Mathematical Structures: Theory and Applications

Permutations Discrete Mathematical Structures: Theory and Applications

Permutations Discrete Mathematical Structures: Theory and Applications

Combinations Discrete Mathematical Structures: Theory and Applications

Combinations Discrete Mathematical Structures: Theory and Applications

Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Generating Permutations and Combinations Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Probability Definition 7.8.1 A probabilistic experiment, or random experiment, or simply an experiment, is the process by which an observation is made. In probability theory, any action or process that leads to an observation is referred to as an experiment. Examples include: Tossing a pair of fair coins. Throwing a balanced die. Counting cars that drive past a toll booth. Discrete Mathematical Structures: Theory and Applications

Discrete Probability Definition 7.8.3 The sample space associated with a probabilistic experiment is the set consisting of all possible outcomes of the experiment and is denoted by S. The elements of the sample space are referred to as sample points. A discrete sample space is one that contains either a finite or a countable number of distinct sample points. Discrete Mathematical Structures: Theory and Applications

Discrete Probability Definition 7.8.6 Definition 7.8.7 An event in a discrete sample space S is a collection of sample points, i.e., any subset of S. In other words, an event is a set consisting of possible outcomes of the experiment. Definition 7.8.7 A simple event is an event that cannot be decomposed. Each simple event corresponds to one and only one sample point. Any event that can be decomposed into more than one simple event is called a compound event. Discrete Mathematical Structures: Theory and Applications

Discrete Probability Definition 7.8.8 Let A be an event connected with a probabilistic experiment E and let S be the sample space of E. The event B of nonoccurrence of A is called the complementary event of A. This means that the subset B is the complement A’ of A in S. In an experiment, two or more events are said to be equally likely if, after taking into consideration all relevant evidences, none can be expected in reference to another. Discrete Mathematical Structures: Theory and Applications

Discrete Probability Discrete Mathematical Structures: Theory and Applications

Discrete Probability Axiomatic Approach Analyzing the concept of equally likely probability, we see that three conditions must hold. The probability of occurrence of any event must be greater than or equal to 0. The probability of the whole sample space must be 1. If two events are mutually exclusive, the probability of their union is the sum of their respective probabilities. These three fundamental concepts form the basis of the definition of probability. Discrete Mathematical Structures: Theory and Applications

Discrete Probability Discrete Mathematical Structures: Theory and Applications

Discrete Probability Discrete Mathematical Structures: Theory and Applications

Discrete Probability Discrete Mathematical Structures: Theory and Applications

Discrete Probability Conditional Probability Consider the throw of two distinct balanced dice. To find the probability of getting a sum of 7, when it is given that the digit in the first die is greater than that in the second. In the probabilistic experiment of throwing two dice the sample space S consists of 6 * 6 = 36 outcomes. Assume that each of these outcomes is equally likely. Let A be the event: The sum of the digits of the two dice is 7, and let B be the event: The digit in the first die is greater than the second. Discrete Mathematical Structures: Theory and Applications

Discrete Probability Conditional Probability P(A)=6/36 B : {(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (5 , 1), (5 , 2), (5 , 3),(5 , 4), (4, 1), (4, 2), (4, 3), (3, 1), (3, 2), (2, 1)}. P(B)= 15/36=0.417 Discrete Mathematical Structures: Theory and Applications

Discrete Probability Conditional Probability Let C be the event: The sum of the digits in the two dice is 7 but the digit in the first die is greater than the second. Then C : {(6, 1), (5 , 2), (4, 3)} = A ∩ B. P(A ∩ B)=3/36=0.083 P(A|B)=0.083/0.417=0.199 Discrete Mathematical Structures: Theory and Applications