STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia


 Coral Wilkinson
 2 years ago
 Views:
Transcription
1 STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that an event will happen can be described by a number between 0 and 1: A probability of 0, or 0%, means the event has no chance of happening. A probability of 1/2, or 50%, means the event is just as likely to happen as not to happen. A probability of 1, or 100%, means the event is certain to happen. For instance, the probability of a coin landing heads up is ½, or 50%, This means you would expect a coin to land heads up half of the time. 1
2 2.1  Sample Space The sample space of a statistical experiment, denoted by S, is the set of all possible outcomes of that experiment. Ex. Roll a die Outcomes: landing with a 1, 2, 3, 4, 5, or 6 face up. Sample Space: S {1, 2, 3, 4, 5, 6} Sample Space Example 2 : A automobile consultant records fuel type and vehicle type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV The Sample Space: e 1 e 1 e 2 e 3 e 4 e 5 e 6 Gasoline, Truck Gasoline, Car Gasoline, SUV Diesel, Truck Diesel, Car Diesel, SUV Car Car e 2 e 3 e 4 e 5 e 6 2
3 Sample Space Example 3: Suppose that three items are selected at random from a manufacturing process. Each item is inspected and classified defective (D) or nondefective (N). To list the elements of the sample space, we construct the tree diagram. Sample Space: S {DDD, DDN, DND, DNN, NDD, NDN, NND, NNN} 2.2 Events An event is any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome. Example 1: We may be interested in the event A that the outcome when a die is tossed is divisible by 3. This will occur if the outcome is an element of the subset A {3,6} of the sample space S. 3
4 Events Relations from the Set Theory 1The complement of an event A with respect to S is the subset of all elements of S that are not in A. We denote the complement, of A by the symbol A. 2The intersection of two events A and, denoted by the symbol A I, is the event containing all elements that are common to A and. AI Read: A and A A Events Two events A and are mutually exclusive, or disjoint, if A I φ ie, if A and have no elements in common. 3 The union of the two events A and, denoted by the symbol AU, is the event containing all the elements that belong to A or or both. Read A or AU Mutually Exclusive A A 4
5 Venn Diagrams AU AI A A Mutually Exclusive A A Events Example 1 : Rolling a die. S {1, 2, 3, 4, 5, 6} Let A {1, 2, 3} and {1, 3, 5} A U {1,2,3,5} A I {1,3} {, A {4,5,6} 5
6 Events Example 3: In a Venn diagram we let the sample space be a rectangle and represent events by circles drawn inside the rectangle. EXERCISE 2.1 List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8. (b) the set S {x x2 + 4x  5 0}; (c) the set of outcomes when a coin is tossed until a tail or three heads appear. (d) the set S {x a; is a continent}; (e) the set. S {x \ 2x  4 > 0 and X < 1}. 6
7 Solution 2.1 (a) S {8, 16, 24, 32, 40, 48}. (b) For x 2 + 4x 5 (x + 5)(x 1) 0, the solutions are: x 55 and x 1. So, the sample space S { 5, 1}. (c) S {T,HT,HHT,HHH}. (d) S {N. America, S. America, Europe, Asia, Africa, Australia, Antarctica}. (e) Solving 2x 4 0 gives x 2. Since we must also have x < 1, it follows that S φ Exercise 2 (2.14) Let S {0,1,2,3,4,5,6,7,8,9} and A {0,2,4,6,8}, {1,3,5,7,9}, C {2,3,4,5}, and D {1,6, 7}, List the elements of the sets corresponding to the following events: 7
8 Solution robability of an Event 8
9 2.4  robability of an Event The probability of an event A corresponds to the occurrence of that event; It is characterized by: If the events A1, A2, A3, are mutually exclusive events, then : Example: A coin is tossed twice (2 times). What is the probability that at least one head occurs? Solution: The sample space; for this experiment is: If the coin is balanced, each of these outcomes would be equally likely to occur. Therefore, we assign a probability of w to each sample point. Then 4w 1, or w 1/4. If A represents the event of at least one1 head occurring, then A {HH, HT, TH} and ( A)
10 Example: let A be the event that an even number turns up and let be the event, that, a number divisible by 3 occurs. Find (A U ) and ( ) A I Solution: For the events A {2,4,6} and {3,6} we have y assigning a probability of 1/9 to each odd number and 2/9 to each even number, we have ( A ) ( A ) robability of an Event Theorem: If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, then the probability of event A is ( A) n N 10
11 Example: A statistics class for engineers consists of 25 industrial, 10 mechanical, 10 electrical, and 8 civil engineering students. If a person is randomly selected by the instructor to answer a question, find the probability that the student chosen is (a) an industrial engineering major, (b) a civil engineering or an electrical engineering major. Solution: Denote by : M, E, and C the students majoring in industrial, mechanical, electrical, and civil engineering, respectively. The total number of students in the class is 53. all of which arc equally likely to be selected. Since 25 of the 53 students are majoring in industrial engineering, the probability of event: ( I ) Since 18 of the 53 students are civil or electrical engineering majors, it follows that: 18 ( C U E ) Additive Rules Theorem 1: If A and are two events, then ( A U ) ( A ) + ( ) ( A ) Corollary 1: If A and are mutually exclusive, then ( AU ) ( A) + ( ) If A and are mutually exclusive, then ( A I ) 0. 11
12 Additive Rules Corollary 2: If A 1, A 2,..A n are mutually exclusive, then ( A1 U A2... An ) ( A1 ) + ( A2 )... + ( An ) Theorem 2: If A and A are complementary events, then ( A) + ( A') 1 Example: A card is drawn from a wellshuffled deck of 52 playing cards. What is the probability that it is a queen or a heart? Solution: Q Queen and H Heart ( Q ), ( H ), ( Q I H ) Q ( U H ) Q ( ) + H ( ) Q ( I H )
13 Conditional robability For any two events A and with () > 0, the conditional probability bilit of A given that t has occurred is defined d by ( ) A ( ) ( ) A Which can be written: ( ) ( ) ( ) A A Example: Example: Consider the toss of two dice. Let E {sum of spots on dice is 4} F {sum of spots on dice is at most 4}. (E) 1/12 since E {(1, 3), (2, 2), (3, 1)}. (F) 1/6 since F {(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1)}. What about (E F)? E {sum of spots on dice is 4} F {sum of spots on dice is at most 4} ( E F) ( E F) ( F) ( E) ( F)
14 Independence Two events A and are independent events if Or ( A ) ( A). ( / A) ( A) Otherwise A and are dependent. Multiplicative Rule If in an experiment the events A and can both occur, then ( A ) ( A) ( / A) 14
15 Multiplicative Rule Events A and are independent events if and only if ( A ) A ( ) ( ) Note: this generalizes for more than two independent events. Example One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first bag and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black? 15
16 Thank You Any Questions? STAT 319 robability and Statistics For Engineers Dr Mohamed AICHOUNI & Dr Mustapha OUKENDAKDJI
A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.
Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome
More informationProbability. Experiment  any happening for which the result is uncertain. Outcome the possible result of the experiment
Probability Definitions: Experiment  any happening for which the result is uncertain Outcome the possible result of the experiment Sample space the set of all possible outcomes of the experiment Event
More information7.5 Conditional Probability; Independent Events
7.5 Conditional Probability; Independent Events Conditional Probability Example 1. Suppose there are two boxes, A and B containing some red and blue stones. The following table gives the number of stones
More information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
More informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics
More informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationAn event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event
An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the
More information+ Section 6.2 and 6.3
Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities
More informationStatistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then
Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance
More informationPROBABILITY NOTIONS. Summary. 1. Random experiment
PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non
More informationLesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationMATH 3070 Introduction to Probability and Statistics Lecture notes Probability
Objectives: MATH 3070 Introduction to Probability and Statistics Lecture notes Probability 1. Learn the basic concepts of probability 2. Learn the basic vocabulary for probability 3. Identify the sample
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationDefinition Sample Space  The collection of all possible outcomes of a chance experiment is the sample space for the experiment.
Probability We will discuss different aspects of probability, from its definition to the various rules associated with probability. From independent events to disjoint events to events with replacement
More information7 Probability. Copyright Cengage Learning. All rights reserved.
7 Probability Copyright Cengage Learning. All rights reserved. 7.1 Sample Spaces and Events Copyright Cengage Learning. All rights reserved. Sample Spaces 3 Sample Spaces At the beginning of a football
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationMath 117 Chapter 7 Sets and Probability
Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a welldefined collection of specific objects. Each item in the set is called an element or a member. Curly
More informationnumber of equally likely " desired " outcomes numberof " successes " OR
Math 107 Probability and Experiments Events or Outcomes in a Sample Space: Probability: Notation: P(event occurring) = numberof waystheevent canoccur total number of equally likely outcomes number of equally
More informationMATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics
MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should
More informationToss a coin twice. Let Y denote the number of heads.
! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.
More informationProbability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.
1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event
More informationChapter Chapter Goals. Assessing Probability. Important Terms. Events. Sample Space. Chapter 4 Basic Probability
Chapter 4 4 Chapter Goals Chapter 4 Basic Probability fter completing this chapter, you should be able to: Explain basic probability concepts and definitions Use contingency tables to view a sample space
More informationThe study of probability has increased in popularity over the years because of its wide range of practical applications.
6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,
More informationEvents. Independence. Coin Tossing. Random Phenomena
Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,
More informationExample: If we roll a dice and flip a coin, how many outcomes are possible?
12.5 Tree Diagrams Sample space Sample point Counting principle Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible
More informationBasic Probability Theory I
A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population
More informationPROBABILITY. Chapter Overview
Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability
More informationBasic concepts in probability. Sue Gordon
Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are
More informationA Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability  1. Probability and Counting Rules
Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample
More information3.1 Events, Sample Spaces, and Probability
University of California, Davis Department of Statistics Summer Session II Statistics 13 August 6, 2012 Lecture 3: Probability 3.1 Events, Sample Spaces, and Probability Date of latest update: August 8
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical
More information1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.
1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first
More informationAP Stats  Probability Review
AP Stats  Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
More informationBasic Probability Theory II
RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More informationThe random variable X  the no. of defective items when three electronic components are tested would be
RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS Example: Give the sample space giving a detailed description of each possible outcome when three electronic components are tested, where N  denotes nondefective
More informationChapter 4: Probabilities and Proportions
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,
More informationProbability: Events and Probabilities
Probability: Events and Probabilities PROBABILITY: longrun relative frequency; likelihood or chance that an outcome will happen. A probability is a number between 0 and 1, inclusive, EVENT: An outcome
More informationLecture 11: Probability models
Lecture 11: Probability models Probability is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model we need the following ingredients A sample
More informationChapter. Probability Pearson Education, Inc. All rights reserved. 1 of 20
Chapter 3 Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 20 Chapter Outline 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition
More informationMAT 1000. Mathematics in Today's World
MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities
More informationMath 421: Probability and Statistics I Note Set 2
Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationProbability Review. ICPSR Applied Bayesian Modeling
Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How
More information36 Odds, Expected Value, and Conditional Probability
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
More informationProbability. Vocabulary
MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability
More information7.1 Sample space, events, probability
7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationExam 1 Review Math 118 All Sections
Exam Review Math 8 All Sections This exam will cover sections..6 and 2.2.3 of the textbook. No books, notes, calculators or other aids are allowed on this exam. There is no time limit. It will consist
More informationBasics of Probability
Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.
More informationProbability and Counting
Probability and Counting Basic Counting Principles Permutations and Combinations Sample Spaces, Events, Probability Union, Intersection, Complements; Odds Conditional Probability, Independence Bayes Formula
More informationPROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE
PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch.  Problems to look at Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
More informationChingHan Hsu, BMES, National Tsing Hua University c 2014 by ChingHan Hsu, Ph.D., BMIR Lab
Lecture 2 Probability BMIR Lecture Series in Probability and Statistics ChingHan Hsu, BMES, National Tsing Hua University c 2014 by ChingHan Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random
More informationMath 1320 Chapter Seven Pack. Section 7.1 Sample Spaces and Events. Experiments, Outcomes, and Sample Spaces. Events. Complement of an Event
Math 1320 Chapter Seven Pack Section 7.1 Sample Spaces and Events Experiments, Outcomes, and Sample Spaces An experiment is an occurrence with a result, or outcome, that is uncertain before the experiment
More information7.5: Conditional Probability
7.5: Conditional Probability Example 1: A survey is done of people making purchases at a gas station: buy drink (D) no drink (Dc) Total Buy drink(d) No drink(d c ) Total Buy Gas (G) 20 15 35 No Gas (G
More informationProbability and Venn diagrams UNCORRECTED PAGE PROOFS
Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve
More informationMath 150 Sample Exam #2
Problem 1. (16 points) TRUE or FALSE. a. 3 die are rolled, there are 1 possible outcomes. b. If two events are complementary, then they are mutually exclusive events. c. If A and B are two independent
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationLesson 3 Chapter 2: Introduction to Probability
Lesson 3 Chapter 2: Introduction to Probability Department of Statistics The Pennsylvania State University 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of
More informationLecture 2: Probability
Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability
More informationPROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA
PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet
More informationRemember to leave your answers as unreduced fractions.
Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,
More informationACMS 10140 Section 02 Elements of Statistics October 28, 2010. Midterm Examination II
ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages of the examination
More informationProbability OPRE 6301
Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.
More informationSection 8.1 Properties of Probability
Section 8. Properties of Probability Section 8. Properties of Probability A probability is a function that assigns a value between 0 and to an event, describing the likelihood of that event happening.
More informationLaws of probability. Information sheet. Mutually exclusive events
Laws of probability In this activity you will use the laws of probability to solve problems involving mutually exclusive and independent events. You will also use probability tree diagrams to help you
More informationOdds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes.
MATH 11008: Odds and Expected Value Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes. Suppose all outcomes in a sample space are equally likely where a of them
More informationSTAT 270 Probability Basics
STAT 270 Probability Basics Richard Lockhart Simon Fraser University Spring 2015 Surrey 1/28 Purposes of These Notes Jargon: experiment, sample space, outcome, event. Set theory ideas and notation: intersection,
More informationChapter 13 & 14  Probability PART
Chapter 13 & 14  Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14  Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
More informationIntroduction to Probability. Experiments. Sample Space. Event. Basic Requirements for Assigning Probabilities. Experiments
Introduction to Probability Experiments These are processes that generate welldefined outcomes Experiments Counting Rules Combinations Permutations Assigning Probabilities Experiment Experimental Outcomes
More informationQuestion: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationSummary of some Rules of Probability with Examples
Summary of some Rules of Probability with Examples CEE 201L. Uncertainty, Design, and Optimization Department of Civil and Environmental Engineering Duke University Henri P. Gavin Spring, 2016 Introduction
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a datagenerating
More informationBinomial random variables
Binomial and Poisson Random Variables Solutions STATUB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance
More informationChapter 5: Probability: What are the Chances? Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation
Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 5 Probability: What Are
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
More informationProbabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I
Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 525 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationChapter 5 A Survey of Probability Concepts
Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible
More informationCh. 13.3: More about Probability
Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the
More information4.5 Finding Probability Using Tree Diagrams and Outcome Tables
4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or
More informationSection 6.2 ~ Basics of Probability. Introduction to Probability and Statistics SPRING 2016
Section 6.2 ~ Basics of Probability Introduction to Probability and Statistics SPRING 2016 Objective After this section you will know how to find probabilities using theoretical and relative frequency
More informationACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers
ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages
More informationfrequency of E sample size
Chapter 4 Probability (Page 1 of 24) 4.1 What is Probability? Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate
More informationSample Space, Events, and PROBABILITY
Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationPROBABILITY. Chapter Overview Conditional Probability
PROBABILITY Chapter. Overview.. Conditional Probability If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the
More informationChapter 5  Probability
Chapter 5  Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set
More informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques
More informationDistributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment
C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationIn this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events.
Lecture#4 Chapter 4: Probability In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. 42 Fundamentals Definitions:
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More information