Events. Independence. Coin Tossing. Random Phenomena

Size: px
Start display at page:

Download "Events. Independence. Coin Tossing. Random Phenomena"

Transcription

1 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, or trial, generates an outcome Something happens on each trial, and we call whatever happens the outcome The Sample Space S set of all possible outcomes of a random phenomena An event consists of a combination of outcomes Subsets of the sample space The probability of an event is its long-run relative frequency Slide 14-1 Slide 14-2 Coin Tossing Independence Relative Frequency In order to think about what happens with combinations of outcomes, it really simplifies things if the individual trials are independent Roughly speaking, this means that the outcome of one trial doesn t influence or change the outcome of another No of Tosses Slide 14-3 Slide 14-4

2 The Law of Large Numbers Probability The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases The idea of the long run is hard to grasp, so the LLN is often misunderstood The common (mis)understanding is that random phenomena are supposed to compensate some for whatever happened in the past This is just not true Thanks to the LLN, we know that relative frequencies settle down in the long run, so we can officially give the name probability to that value Probabilities must be between 0 and 1, inclusive A probability of 0 indicates impossibility A probability of 1 indicates certainty Slide 14-5 Slide 14-6 Examples of Sample Spaces Rolling Two Dice Sample Space Toss a coin twice S = { HH, HT, TH, TT } Roll a pair of die and record numbers S = {(1,1),(1,2),,(1,6),(2,1),, (2,6),,(6,6)} Roll a pair of die and record total score S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Toss a coin until first tail appears S = {T, HT, HHT HHHT, } Measure duration of charge of mobile phone battery S = { t t 0 } First Die Second Die Slide 14-7 Slide 14-8

3 Events, Combining Events Example: Rolling a die Events uppercase letters, A,, C, Special events: S = certain event = null, impossible event = { } Complement of event A is Ā = all outcomes not in A A Union of events; A or or both A Intersection of events A and Disjoint events cannot occur together, ie A = A = score on die is even = { } = score on die is odd = { } C = score is greater than 4 = { } A = A = A C = C = (A C) ( C) = (A ) C = Slide 14-9 Slide Example: Rolling two dice Probability Distributions A = score on 1 st die is even = { } = score on 2 nd die is odd = { } C = score is greater than 9 = { } A = A = A C = C = (A C) ( C) = (A ) C = Sample space S = {s 1, s 2, s 3, } Probabilities: numbers p 1, p 2, p 3, All p i s lie between 0 and 1 ( 0 p i 1 ) Sum of all p i s is 1: ( p 1 + p 2 + p 3 + = 1 ) Probability of an event obtained by adding up probabilities of all outcomes in A Slide Slide 14-12

4 Equally Likely outcomes Formal Probability Example: Draw a card from a well shuffled pack A = event of drawing an Ace 4 P(A) = Generally P(A) = Number of outcomes in A Number of outcomes in S 1 Two requirements for a probability: A probability is a number between 0 and 1 For any event A, 0 P(A) 1 2 Something has to happen rule : The probability of the set of all possible outcomes of a trial must be 1 P(S) = 1 (S represents the set of all possible outcomes) Slide Slide Formal Probability (cont) Formal Probability (cont) 3 Complement Rule: Definition: The set of outcomes that are not in the event A is called the complement of A, denoted A C, or Ā The probability of an event occurring is 1 minus the probability that it doesn t occur P(A) = 1 P(A C ) 4 Addition Rule: Definition: Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint For two disjoint events A and, the probability that one or the other occurs is the sum of the probabilities of the two events P(A ) = P(A or ) = P(A) + P(), provided that A and are disjoint Slide Slide 14-16

5 Formal Probability (cont) 5 Multiplication Rule: For two independent events A and, the probability that both A and occur is the product of the probabilities of the two events P(A and ) = P(A ) = P(A) x P(), provided that A and are independent Draw a card: event A an Ace; event a heart 1 A Ā Slide Slide What Can Go Wrong? A Ā P(A ) P(Ā ) P() P(A ) P(Ā ) P() P(A) P(Ā) 100 eware of probabilities that don t add up to 1 Don t add probabilities of events if they re not disjoint Don t multiply probabilities of events if they re not independent Don t confuse disjoint and independent disjoint events can t be independent Slide Slide 14-20

MAT 1000. Mathematics in Today's World

MAT 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 information

Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Statistical 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 information

1. 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. 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 information

7 Probability. Copyright Cengage Learning. All rights reserved.

7 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 information

+ Section 6.2 and 6.3

+ 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 information

Basic Probability Theory I

Basic 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 information

Basic concepts in probability. Sue Gordon

Basic 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 information

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY. 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 information

Section 6.2 Definition of Probability

Section 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 information

6.3 Conditional Probability and Independence

6.3 Conditional Probability and Independence 222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

Discrete 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 information

Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

Probability. 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 information

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

Discrete 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 information

I. WHAT IS PROBABILITY?

I. 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 information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card 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 information

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

STAT 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 information

Math/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 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 information

number of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.

number 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 information

Probabilistic Strategies: Solutions

Probabilistic Strategies: Solutions Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

More information

The study of probability has increased in popularity over the years because of its wide range of practical applications.

The 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 information

Chapter 4 Probability

Chapter 4 Probability The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability?

More information

Basic Probability Concepts

Basic 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 information

In the situations that we will encounter, we may generally calculate the probability of an event

In 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 information

Probability definitions

Probability 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 data-generating

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical 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 information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

Discrete 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 information

Probability: 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 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 information

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

More information

Session 8 Probability

Session 8 Probability Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

More information

AP Stats - Probability Review

AP 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 information

Chapter 13 & 14 - Probability PART

Chapter 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 information

Unit 19: Probability Models

Unit 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 information

Section 6-5 Sample Spaces and Probability

Section 6-5 Sample Spaces and Probability 492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

More information

Lecture 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 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 information

PROBABILITY 14.3. section. The Probability of an Event

PROBABILITY 14.3. section. The Probability of an Event 4.3 Probability (4-3) 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 information

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either

More information

The Central Limit Theorem Part 1

The Central Limit Theorem Part 1 The Central Limit Theorem Part. Introduction: Let s pose the following question. Imagine you were to flip 400 coins. To each coin flip assign if the outcome is heads and 0 if the outcome is tails. Question:

More information

Introduction to Probability

Introduction 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 information

Basic Probability Theory II

Basic 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 information

Probability OPRE 6301

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 information

Elements of probability theory

Elements 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 information

Chapter 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? 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 information

Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A.

Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. Complement If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. For example, if A is the event UNC wins at least 5 football games, then A c is the

More information

Lecture 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 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 information

Combinatorics: The Fine Art of Counting

Combinatorics: 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 information

36 Odds, Expected Value, and Conditional Probability

36 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 information

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 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 information

Probability: 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 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 information

Chapter 4: Probability and Counting Rules

Chapter 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 information

Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014 Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

More information

Probability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.

Probability. 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 information

E3: PROBABILITY AND STATISTICS lecture notes

E3: 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 information

Probability and Hypothesis Testing

Probability and Hypothesis Testing B. Weaver (3-Oct-25) Probability & Hypothesis Testing. PROBABILITY AND INFERENCE Probability and Hypothesis Testing The area of descriptive statistics is concerned with meaningful and efficient ways of

More information

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015. Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

More information

Elementary probability

Elementary probability Elementary probability Many of the principal applications of calculus are to questions of probability and statistics. We shall include here an introduction to elementary probability, and eventually some

More information

Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)

Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179) Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities

More information

Welcome to Stochastic Processes 1. Welcome to Aalborg University No. 1 of 31

Welcome to Stochastic Processes 1. Welcome to Aalborg University No. 1 of 31 Welcome to Stochastic Processes 1 Welcome to Aalborg University No. 1 of 31 Welcome to Aalborg University No. 2 of 31 Course Plan Part 1: Probability concepts, random variables and random processes Lecturer:

More information

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 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 information

PROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE

PROBABILITY. 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 information

Probability: The Study of Randomness Randomness and Probability Models

Probability: The Study of Randomness Randomness and Probability Models Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4.1 and 4.2 2009 W.H. Freeman and Company Objectives (IPS Chapters 4.1 and 4.2) Randomness and Probability models Probability

More information

ACMS 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 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 information

Introduction to probability theory in the Discrete Mathematics course

Introduction to probability theory in the Discrete Mathematics course Introduction to probability theory in the Discrete Mathematics course Jiří Matoušek (KAM MFF UK) Version: Oct/18/2013 Introduction This detailed syllabus contains definitions, statements of the main results

More information

Concepts of Probability

Concepts of Probability Concepts of Probability Trial question: we are given a die. How can we determine the probability that any given throw results in a six? Try doing many tosses: Plot cumulative proportion of sixes Also look

More information

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

IAM 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 information

Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13. Understanding Probability and Long-Term Expectations Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

More information

The Casino Lab STATION 1: CRAPS

The Casino Lab STATION 1: CRAPS The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

More information

Topic 1 Probability spaces

Topic 1 Probability spaces CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in five-card poker is about 0.2%,

More information

ACMS 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 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 information

3. Discrete Probability. CSE 312 Autumn 2011 W.L. Ruzzo

3. Discrete Probability. CSE 312 Autumn 2011 W.L. Ruzzo 3. Discrete Probability CSE 312 Autumn 2011 W.L. Ruzzo sample spaces Sample space: S is the set of all possible outcomes of an experiment (Ω in your text book Greek uppercase omega) Coin flip: S = {Heads,

More information

Probability. 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) 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 information

Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary 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 information

4: Probability. What is probability? Random variables (RVs)

4: Probability. What is probability? Random variables (RVs) 4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

More information

Math 3C Homework 3 Solutions

Math 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 information

Probability and statistical hypothesis testing. Holger Diessel holger.diessel@uni-jena.de

Probability and statistical hypothesis testing. Holger Diessel holger.diessel@uni-jena.de Probability and statistical hypothesis testing Holger Diessel holger.diessel@uni-jena.de Probability Two reasons why probability is important for the analysis of linguistic data: Joint and conditional

More information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

More information

MATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker)

MATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker) Last modified: February, 00 References: MATHEMATICS 5, SPRING 00 PROBABILITY THEORY Outline # (Combinatorics, bridge, poker) PRP(Probability and Random Processes, by Grimmett and Stirzaker), Section.7.

More information

A Few Basics of Probability

A Few Basics of Probability A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

More information

Statistics 100A Homework 2 Solutions

Statistics 100A Homework 2 Solutions Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

More information

The Calculus of Probability

The Calculus of Probability The Calculus of Probability Let A and B be events in a sample space S. Partition rule: P(A) = P(A B) + P(A B ) Example: Roll a pair of fair dice P(Total of 10) = P(Total of 10 and double) + P(Total of

More information

ECE302 Spring 2006 HW1 Solutions January 16, 2006 1

ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics

More information

4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables 4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

More information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability 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 information

Definition and Calculus of Probability

Definition 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

Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

Decision Making Under Uncertainty. Professor Peter Cramton Economics 300 Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate

More information

Remember to leave your answers as unreduced fractions.

Remember 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 information

Unit 18: Introduction to Probability

Unit 18: Introduction to Probability Unit 18: Introduction to Probability Summary of Video There are lots of times in everyday life when we want to predict something in the future. Rather than just guessing, probability is the mathematical

More information

Math 141. Lecture 3: The Binomial Distribution. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141. Lecture 3: The Binomial Distribution. Albyn Jones 1. 1 Library 304.  jones/courses/141 Math 141 Lecture 3: The Binomial Distribution Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Coin Tossing Coin Tosses Independent Coin Tosses Crucial Features

More information

Math 55: Discrete Mathematics

Math 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 information

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8. Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

More information

ELEMENTARY PROBABILITY

ELEMENTARY 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 information

Ch. 13.3: More about Probability

Ch. 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

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

More information

Binomial random variables

Binomial random variables Binomial and Poisson Random Variables Solutions STAT-UB.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 information

The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

More information

Math Review Large Print (18 point) Edition Chapter 4: Data Analysis

Math Review Large Print (18 point) Edition Chapter 4: Data Analysis GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter 4: Data Analysis Copyright 2010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,

More information

Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability. 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 information

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values. MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

More information

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Math 1342 (Elementary Statistics) Test 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the indicated probability. 1) If you flip a coin

More information

Probabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I

Probabilities. 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 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with

More information

Chapter 4 - Practice Problems 2

Chapter 4 - Practice Problems 2 Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the

More information