Introduction to Probability. Experiments. Sample Space. Event. Basic Requirements for Assigning Probabilities. Experiments


 Arthur Daniel
 2 years ago
 Views:
Transcription
1 Introduction to Probability Experiments These are processes that generate welldefined outcomes Experiments Counting Rules Combinations Permutations Assigning Probabilities Experiment Experimental Outcomes Toss a coin Head, tail Select a part for Defective, nondefective inspection Conduct a sales call Purchase, no purchase Roll a die 1,, 3,, 5, 6 Play a football game Win, lose, tie Sample Space Event The sample space for an experiment is the set of all experimental outcomes Any subset of the sample space is called an event. S Head,Tail For a coin toss: Rolling a die: S 1,, 3,, 5, 6 Selecting a part for inspection: S 1,, 3,, 5, 6 Rolling a die: S Defective, Nondefective Events: { 1 } // the outcome is 1 (elementary event) { 1, 3, 5 } // the outcome is an odd number {, 5, 6 } // the outcome is at least. Probability is a numerical measure of the likelihood of an event occurring Basic Requirements for Assigning Probabilities Let E i denote the ith experimental outcome (elementary event) and P(E i ) is its probability of occurring. Then: 0 P( E i ) 1for all i Probability: The sum of the probabilities for all experimental outcomes must be must equal 1. For n experimental outcomes: P( E1) P( E)... P( En) 1 The occurrence of the event is just as likely as it is unlikely 1
2 Principle of Indifference We assign equal probability to elementary events if we have no reason to expect one over the other. 1 P( E i ) n For a coin toss: S Head, Tail P(Head) = P(Tail) = 1/ Rolling a die: S 1,, 3,, 5, 6 P(1) = P() = = P(6) = 1/6 Selecting a part for inspection: S Defective,Nondefective P(Defective) =? This method of assigning probabilities is indicated if each experimental outcome is equally likely Relative Frequency Method This method is indicated when the data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times. What if experimental outcomes are NOT equally likely. Then the Principle of Indifference is out. We must assign probabilities on the basis of experimentation or historical data. Selecting a part S Defective,Nondefective for inspection: N parts: n 1 defective and n nondefective P(Defective) = n 1 /N, P(Nondefective) = n /N Counting Experimental Outcomes To assign probabilities, we must first count experimental outcomes. We have 3 useful counting rules for multiplestep experiments. For example, what is the number of possible outcomes if we roll the die times? 1. Counting rule for multistep experiments. Counting rule for combinations 3. Counting rule for permutations Example: Lucas Tool Rental Relative Frequency Method Ace Rental would like to assign probabilities to the number of carpet cleaners it rents each day. Office records show the following frequencies of daily rentals for the last 0 days. Number of Cleaners Rented Number of Days Relative Frequency Method Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days). Number of Cleaners Rented Number of Days Probability /0 Subjective Method When economic conditions and a company s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data. We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur. The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate.
3 Counting Rule for Multi Step Experiments If an experiment can be described as a sequence of k steps with n 1 possible outcomes on the first step, n possible outcomes on the second step, then the total number of experimental outcomes is given by: ( n1 )( n)...( nk ) Example: Bradley Investments Bradley has invested in two stocks, Markley Oil and Collins Mining. Bradley has determined that the possible outcomes of these investments three months from now are as follows. Investment Gain or Loss in 3 Months (in $000) Markley Oil Collins Mining A Counting Rule for Multiple Step Experiments Bradley Investments can be viewed as a two step experiment. It involves two stocks, each with a set of experimental outcomes. Markley Oil: n 1 = Collins Mining: n = Total Number of Experimental Outcomes: n 1 n = ()() = 8 Markley Oil (Stage 1) Gain 10 Lose 0 Tree Diagram Gain 5 Even Collins Mining (Stage ) Gain 8 Lose Gain 8 Lose Gain 8 Lose Gain 8 Lose Experimental Outcomes (10, 8) Gain $18,000 (10, ) Gain $8,000 (5, 8) Gain $13,000 (5, ) Gain $3,000 (0, 8) Gain $8,000 (0, ) Lose $,000 (0, 8) Lose $1,000 (0, ) Lose $,000 where And by definition Counting Rule This forrule allows us to count the number of Combinations experimental outcomes when we select n objects from a (usually larger) set of N objects. The number of N objects taken n at a time is C N n N N! n n!( N n)! N! N( N 1)( N )...()(1) n! n( n 1)( n )...()(1) 0! 1 Example: Quality Control An inspector randomly selects of 5 parts for inspection. In a group of 5 parts, how many combinations of parts can be selected? 5 5! (5)()(3)()(1) 10 C!(5 )! ()(1)(3)()(1) 1 5 Let the parts de designated A, B, C, D, E. Thus we could select: AB AC AD AE BC BD BE CD CE and DE 10 3
4 C Iowa Lottery Iowa randomly selects 6 integers from a group of 7 to determine the weekly winner. What are your odds of winning if you purchased one ticket? 7 7! (7)(6)(5)()(3)() 6 6!(7 6)! (6)(5)()(3)()(1) ,737,573 Counting Rule for Permutations P N n Sometimes the order of selection matters. This rule allows us to count the number of experimental outcomes when n objects are to be selected from a set of N objects and the order of selection matters. N N! n! n ( N n)! Example: Quality Control Again An inspector randomly selects of 5 parts for inspection. In a group of 5 parts, how many permutations of parts can be selected? 5! 5! (5)()(3)()(1) 10 P (5 )! 3! (3)()(1) 6 5 Again let the parts be designated A, B, C, D, E. Thus we could select: AB BA AC CA AD DA AE EA BC CB BD DB BE EB CD DC CE EC DE and ED 0 Some Basic Relationships of Probability There are some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities. Complement of an Event Union of Two Events Intersection of Two Events Mutually Exclusive Events Complement of an Event Union of Two Events The complement of event A is defined to be the event consisting of all sample points that are not in A. The complement of A is denoted by A c. The union of events A and B is the event containing all sample points that are in A or B or both. The union of events A and B is denoted by A B Event A A c Space S Sample Event A Event B Sample Space S Venn Diagram
5 Union of Two Events Intersection of Two Events M C = Markley Oil Profitable or Collins Mining Profitable M C = {(10, 8), (10, ), (5, 8), (5, ), (0, 8), (0, 8)} P(M C) = P(10, 8) + P(10, ) + P(5, 8) + P(5, ) + P(0, 8) + P(0, 8) = =.8 The intersection of events A and B is the set of all sample points that are in both A and B. The intersection of events A and B is denoted by A Event A Event B Intersection of A and B Sample Space S Intersection of Two Events Addition Law M C = Markley Oil Profitable and Collins Mining Profitable M C = {(10, 8), (5, 8)} P(M C) = P(10, 8) + P(5, 8) = =.36 The addition law provides a way to compute the probability of event A, or B, or both A and B occurring. The law is written as: P(A B) = P(A) + P(B) P(A B Addition Law Mutually Exclusive Events M C = Markley Oil Profitable or Collins Mining Profitable We know: P(M) =.70, P(C) =.8, P(M C) =.36 Thus: P(M C) = P(M) + P(C) P(M C) = =.8 (This result is the same as that obtained earlier using the definition of the probability of an event.) Two events are said to be mutually exclusive if the events have no sample points in common. Two events are mutually exclusive if, when one event occurs, the other cannot occur. Event A Event B Sample Space S 5
6 Mutually Exclusive Events Conditional Probability If events A and B are mutually exclusive, P(A B = 0. The addition law for mutually exclusive events is: P(A B) = P(A) + P(B) The probability of an event given that another event has occurred is called a conditional probability. The conditional probability of A given B is denoted by P(A B). A conditional probability is computed as follows : there s no need to include P(A B PA ( B PA ( B) ) PB ( ) Conditional Probability Multiplication Law PC ( M ) = Collins Mining Profitable given Markley Oil Profitable We know: P(M C) =.36, P(M) =.70 PC ( M ).36 Thus: PC ( M ).513 PM ( ).70 The multiplication law provides a way to compute the probability of the intersection of two events. The law is written as: P(A B) = P(B)P(A B) Multiplication Law Independent Events M C = Markley Oil Profitable and Collins Mining Profitable We know: P(M) =.70, P(C M) =.513 Thus: P(M C) = P(M)P(M C) = (.70)(.513) =.36 (This result is the same as that obtained earlier using the definition of the probability of an event.) If the probability of event A is not changed by the existence of event B, we would say that events A and B are independent. Two events A and B are independent if: P(A B) = P(A) or P(B A) = P(B) 6
7 Multiplication Law for Independent Events The multiplication law also can be used as a test to see if two events are independent. The law is written as: P(A B) = P(A)P(B) Multiplication Law for Independent Events Are events M and C independent? DoesP(M C) = P(M)P(C)? We know: P(M C) =.36, P(M) =.70, P(C) =.8 But: P(M)P(C) = (.70)(.8) =.3, not.36 Hence: M and C are not independent. Terminology Events may or may not be mutually exclusive. If E and F are mutually exclusive events, then P(E U F) = P(E) + P(F) If E and F are not mutually exclusive, then P(E U F) = P(E) + P(F) P(E n F). All elementary events are mutually exclusive. The birth of a son or a daughter are mutually exclusive events. The event that the outcome of rolling a die is even and the event that the outcome of rolling a die is at least four are not mutually exclusive. Simple probabilities If A and B are mutually exclusive events, then the probability of either A or B to occur is the union P(A B) P(A) P(B) Example: The probability of a hat being red is ¼, the probability of the hat being green is ¼, and the probability of the hat being black is ½. Then, the probability of a hat being red OR black is ¾. Simple probabilities If A and B are independent events, then the probability that both A and B occur is the intersection P(A B) P(A) P(B) 7
8 Simple probabilities Example: The probability that a US president is bearded is ~1%, the probability that a US president died in office is ~19%. If the two events are independent, the probability that a president both had a beard and died in office is ~3%. In reality, bearded presidents died in office. (A close enough result.) Harrison, Taylor, Lincoln*, Garfield*, McKinley*, Harding, Roosevelt, Kennedy* (*assassinated) Conditional probabilities What is the probability of event A to occur given that event B did occur. The conditional probability of A given B is P(A B) P(A B) P(A) Example: The probability that a US president dies in office if he is bearded 0.03/0.1 = %. Thus, out of 6 bearded presidents, % are expected to die in office. In reality, died. (Again, a close enough result.) Probability Distribution The probability distribution refers to the frequency with which all possible outcomes occur. There are numerous types of probability distribution. The Uniform Distribution A variable is said to be uniformly distributed if the probability of all possible outcomes are equal to one another. Thus, the probability P(i), where i is one of n possible outcomes, is P(i) 1 n The Binomial Distribution A process that has only two possible outcomes is called a binomial process. In statistics, the two outcomes are frequently denoted as success and failure. The probabilities of a success or a failure are denoted by p and q, respectively. Note that p + q = 1. The binomial distribution gives the probability of exactly k successes in n trials P(k) n p k 1 p n k k The Binomial Distribution The mean and variance of a binomially distributed variable are given by np V npq 8
9 The Poisson distribution Siméon Denis Poisson Poisson d April The Poisson distribution When the probability of success is very small, e.g., the probability of a mutation, then p k and (1 p) n k become too small to calculate exactly by the binomial distribution. In such cases, the Poisson distribution becomes useful. Let be the expected number of successes in a process consisting of n trials, i.e., = np. The probability of observing k successes is P(k) k e The mean and variance of a Poisson distributed variable are given by = and V =, respectively. k! Normal Distribution Gamma Distribution 9
Probability 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 informationPROBABILITIES AND PROBABILITY DISTRIBUTIONS
Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL
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 informationIntroduction to Probability
3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible 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 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 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. 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 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 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 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 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 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 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 informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
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
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 informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More informationChapter 3: The basic concepts of probability
Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording
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 informationP (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or nondeterministic experiments. Suppose an experiment can be repeated any number of times, so that we
More informationProbability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)
Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability
More informationUnit 4 The Bernoulli and Binomial Distributions
PubHlth 540 4. Bernoulli and Binomial Page 1 of 19 Unit 4 The Bernoulli and Binomial Distributions Topic 1. Review What is a Discrete Probability Distribution... 2. Statistical Expectation.. 3. The Population
More informationCONTINGENCY (CROSS TABULATION) TABLES
CONTINGENCY (CROSS TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods
More informationProbability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.
Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how
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 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 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 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 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 informationChapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams
Review for Final Chapter 2: Data quantifiers: sample mean, sample variance, sample standard deviation Quartiles, percentiles, median, interquartile range Dot diagrams Histogram Boxplots Chapter 3: Set
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 informationCh5: Discrete Probability Distributions Section 51: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 51: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationJan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 5054)
Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0 Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample
More informationWorked examples Basic Concepts of Probability Theory
Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
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 informationPROBABILITY. Chapter. 0009T_c04_133192.qxd 06/03/03 19:53 Page 133
0009T_c04_133192.qxd 06/03/03 19:53 Page 133 Chapter 4 PROBABILITY Please stand up in front of the class and give your oral report on describing data using statistical methods. Does this request to speak
More informationUnit 19: Probability Models
Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,
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 informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationSTA 371G: Statistics and Modeling
STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g
More information4. Binomial Expansions
4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
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 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 information108 Combinations and and Permutations. Holt Algebra 1 1
108 Combinations and and Permutations 1 Warm Up For a main dish, you can choose steak or chicken; your side dish can be rice or potatoes; and your drink can be tea or water. Make a tree diagram to show
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 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 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 informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli
More informationOnline EFFECTIVE AS OF JANUARY 2013
2013 A and C Session Start Dates (AB Quarter Sequence*) 2013 B and D Session Start Dates (BA Quarter Sequence*) Quarter 5 2012 1205A&C Begins November 5, 2012 1205A Ends December 9, 2012 Session Break
More informationChapter 4 Probability
The Big Picture of Statistics Chapter 4 Probability Section 42: Fundamentals Section 43: Addition Rule Sections 44, 45: Multiplication Rule Section 47: Counting (next time) 2 What is probability?
More informationIn the situations that we will encounter, we may generally calculate the probability of an event
What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead
More informationProbability & Probability Distributions
Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions
More informationStudy Manual for Exam P/Exam 1. Probability
Study Manual for Exam P/Exam 1 Probability Eleventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com
More informationProbability and Hypothesis Testing
B. Weaver (3Oct25) Probability & Hypothesis Testing. PROBABILITY AND INFERENCE Probability and Hypothesis Testing The area of descriptive statistics is concerned with meaningful and efficient ways of
More informationMathematical goals. Starting points. Materials required. Time needed
Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
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 informationStudy Manual for Exam P/Exam 1. Probability
Study Manual for Exam P/Exam 1 Probability Seventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com
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 informationAlgebra 2 C Chapter 12 Probability and Statistics
Algebra 2 C Chapter 12 Probability and Statistics Section 3 Probability fraction Probability is the ratio that measures the chances of the event occurring For example a coin toss only has 2 equally likely
More informationRandom Variable: A function that assigns numerical values to all the outcomes in the sample space.
STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.
More informationChapter 4. Probability and Probability Distributions
Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the
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 information7 Probability. Copyright Cengage Learning. All rights reserved.
7 Probability Copyright Cengage Learning. All rights reserved. 7.2 Relative Frequency Copyright Cengage Learning. All rights reserved. Suppose you have a coin that you think is not fair and you would like
More information2 Binomial, Poisson, Normal Distribution
2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability
More informationCh. 13.2: Mathematical Expectation
Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we
More informationHomework 6 (due November 4, 2009)
Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More informationSolutions to SelfHelp Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)}
1.4 Basics of Probability 37 Solutions to SelfHelp Exercises 1.3 1. Consider the outcomes as ordered pairs, with the number on the bottom of the red one the first number and the number on the bottom of
More informationDiscrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22
CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair sixsided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than
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 informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant
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 informationVisa Smart Debit/Credit Certificate Authority Public Keys
CHIP AND NEW TECHNOLOGIES Visa Smart Debit/Credit Certificate Authority Public Keys Overview The EMV standard calls for the use of Public Key technology for offline authentication, for aspects of online
More information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More information4. Joint Distributions
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose
More informationPeople have thought about, and defined, probability in different ways. important to note the consequences of the definition:
PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models
More informationExam 2 Study Guide and Review Problems
Exam 2 Study Guide and Review Problems Exam 2 covers chapters 4, 5, and 6. You are allowed to bring one 3x5 note card, front and back, and your graphing calculator. Study tips: Do the review problems below.
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 SECOND EDITION
PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All
More informationPractice Problems #4
Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiplechoice
More information2Probability CHAPTER OUTLINE LEARNING OBJECTIVES
2Probability CHAPTER OUTLINE 21 SAMPLE SPACES AND EVENTS 21.1 Random Experiments 21.2 Sample Spaces 21.3 Events 21.4 Counting Techniques (CD Only) 22 INTERPRETATIONS OF PROBABILITY 22.1 Introduction
More information111 Permutations and Combinations
Fundamental Counting Principal 111 Permutations and Combinations Using the Fundamental Counting Principle 1a. A makeyourownadventure story lets you choose 6 starting points, gives 4 plot choices, and
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 information6.2. Discrete Probability Distributions
6.2. Discrete Probability Distributions Discrete Uniform distribution (diskreetti tasajakauma) A random variable X follows the dicrete uniform distribution on the interval [a, a+1,..., b], if it may attain
More informationCounting principle, permutations, combinations, probabilities
Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing
More informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationTOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities
TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE Roulette is one of the most popular casino games. The name roulette is derived from the French word meaning small
More informationELEMENTARY PROBABILITY
ELEMENTARY PROBABILITY Events and event sets. Consider tossing a die. There are six possible outcomes, which we shall denote by elements of the set {A i ; i =1, 2,...,6}. A numerical value is assigned
More information4.4 Conditional Probability
4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.
More informationElements of probability theory
The role of probability theory in statistics We collect data so as to provide evidentiary support for answers we give to our many questions about the world (and in our particular case, about the business
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 information