# The Story. Probability - I. Plan. Example. A Probability Tree. Draw a probability tree.

Save this PDF as:

Size: px
Start display at page:

Download "The Story. Probability - I. Plan. Example. A Probability Tree. Draw a probability tree."

## Transcription

1 Great Theoretical Ideas In Computer Science Victor Adamchi CS 5-5 Carnegie Mellon University Probability - I The Story The theory of probability was originally developed by a French mathematician Blaise Pascal. Pascal had a friend, fond of gambling, who ased him for strategies. Probability was invented to analyze gambling. This is not Great Theoretical Ideas in Gambling, but in Computer Science. Plan Sample and vents Conditional Probability Bayes Law Law of Total Probability Use of Generating Functions ample Mary flips a fair coin. If it s a head, she rolls a -sided die. If it s a tail, she rolls a -sided die. What is the probability die roll is or higher? Draw a probability tree. Prob: -die A Probability Tree H coin T -die (H,) (H,) (H,) (T,) (T,) (T,) (T,) / / / /8 /8 /8 /8 Label the leaves with outcomes Under each, write its probability: multiply along the path Outcome: A leaf in the probability tree. I.e., a possible sequence of values of all calls to generators in an eecution. Sample Space: The set of all outcomes..g., { (H,), (H,), (H,), (T,), (T,), (T,), (T,) } Probability: ach outcome has a nonnegative probability. Sum of all outcomes probabilities always.

2 ample Mary flips a fair coin. If it s heads, she rolls a -sided die. If it s tails, she rolls a -sided die. What is the probability die roll is or higher? vent: A subset of outcomes. In our eample, = { (H,), (T,), (T,) }. = sum of the probabilities of the outcomes in. coin H T die die (H,) (H,) (H,) (T,) (T,) (T,) (T,) Prob: / / / /8 /8 /8 /8 = roll is or higher = / + /8 + /8 = 5/ We will consider eperiments with a finite number of possible outcomes w, w,..., w n The sample space of the eperiment is the set of all possible outcomes. The roll of a die: = {,,,,5,} ach subset of a sample space is defined to be an event. The event: = {,,} Probability of a event Let X be a random variable which denotes the value of the outcome of a certain eperiment. We will assign probabilities to the possible outcomes of an eperiment. We do this by assigning to each outcome w j a nonnegative number p(w j ) in such a way that p(w ) + + p(w n )= The function p(w j ) is called the probability distribution function of the random variable X, Pr[X = w ] Probabilities For any subset of, we define the probability of to be the number given by event w p(w ) probabilities of outcomes From Random Variables to vents For any random variable X and value a, we can define the event that X = a = Pr[X=a] = Pr[{t X(t)=a}]

3 From Random Variables to vents Random Variable It s a function on the sample space ->[0, ) It s a variable with a probability distribution on its values A coin is tossed ten times. The random variable X is the number of tails that are noted. X can only tae the values 0,,, 0, so X is a discrete random variable. You should be comfortable with both views Theorem The probabilities satisfy the following properties: Pr[ ] = 0 Pr[A B] = Pr[A] + Pr[B] Pr[A B] Pr[A c ] = - Pr[A] Pr[A] = Pr[A B] + Pr[A B c ] B A Uniform Distribution When a coin is tossed and the die is rolled, we assign an equal probability to each outcome. The uniform distribution on a sample space containing n elements is the function p defined by p(w) = /n for every w These should mae sense if we thin of events as sets. ample ample Consider the eperiment that consists of rolling a pair of standard dice. What is the probability of getting a sum of 7 or a sum of? S = { (,), (,), (,), (,), (,5), (,), (,), (,), (,), (,), (,5), (,), (,), (,), (,), (,), (,5), (,), (,), (,), (,), (,), (,5), (,), (5,), (5,), (5,), (5,), (5,5), (5,), (,), (,), (,), (,), (,5), (,) } F We assume that each of outcomes is equally liely. P()= * / P(F)= * / P( F)= 8/

4 A fair coin is tossed 00 times in a row What is the probability that we get eactly half heads? The sample space is the set of all outcomes (sequences) {H,T} 00 ach sequence in is equally liely, and hence has probability / =/ 00 Binomial Distribution = all sequences of 00 tosses Visually t = HHTTT TH Set of all 00 sequences {H,T} 00 vent = Set of sequences with 50 H s and 50 T s P(t) = / S Probability of event = proportion of in S / 00 Birthday Parado How many people do we need to have in a room to mae it a favorable bet (probability of success greater than /) that two people in the room will have the same birthday? Sample space Visually = 5 vent = { w two numbers not the same } ( )( 5 We must find sequences that have no duplication of birthdays. )( 5 ) 5

5 Birthday Problem Pr[in students, some pair share a bday ]= -( )( 5 )( 5 ) 5 Birthday Problem What if there are N possible birthdays? Pr[in m students, some pair share a bday ] -( )( N )( N m ) N For what value of m is this /? m N Cryptographic Hash Functions 99: Rivest publishes MD5. (=8) 99: NSA publishes SHA-0. (=0) 995: NSA publishes SHA-. (=0) SHA- now used in SSL, PGP, 00: NSA also introduces SHA- Birthday Attac Imagine trying to find a collision for SHA-: Tae a huge number of strings, hash them all, hope that two hash to the same 0 bits. This is lie the Birthday Problem with N = 0! So # tries before good chance of collision: Birthday Attac A crypto hash function is considered broen if you can beat the birthday attac. Prof. Xiaoyun Wang ( 王 小 云 ) Infinite Sample Spaces A coin is tossed until the first time that a head turns up. = {,,,, } 005: SHA- collisions in < 9 Later (w/ coauthors): in < SHA- = broen A distribution function: m(n) = -n. Pr w m(w) 5

6 Infinite Sample Spaces Let be the event that the first time a head turns up is after an even number of tosses. = {,,, } Infinite Sample Spaces Suppose your eperiment involves throwing a dart, which is equally liely to land anywhere in the interval [0,]. What is the probability that the dart lands at eactly 0.5? Probability of landing at any rational number is the same If that > 0, then the sum of all probabilities will be >. Thus, the probability that the dart lands at eactly 0.5 is zero. Conditional Probability Consider our voting eample: three candidates A, B, and C are running for office. We decided that A and B have an equal chance of winning and C is only / as liely to win as A. Suppose that before the election is held, A drops out of the race. What are new probabilities to the events B and C P(B A)=/ P(C A)=/ F Conditional Probability Definition. The probability of event F given event is written P(F ) and is defined by P(F ) P(F ) P() We narrow our sample space to. proportion of F to Suppose we roll a white die and blac die What is the probability that the white die is given that the total is 7? S = { (,), (,), (,), (,), (,5), (,), (,), (,), (,), (,), (,5), (,), (,), (,), (,), (,), (,5), (,), (,), (,), (,), (,), (,5), (,), (5,), (5,), (5,), (5,), (5,5), (5,), (,), (,), (,), (,), (,5), (,) } event A = {white die = } event B = {total = 7} Pr[A B] = Pr[A B] A B = = Pr[B] B event A = {white die = } event B = {total = 7}

7 Independence! A and B are independent events if P(A B ) = P(A) P(A B ) = P(A) P(B) P(B A ) = P(B) Silver and Gold One bag has two silver coins, another has two gold coins, and the third has one of each One bag is selected at random. One coin from it is selected at random. It turns out to be gold What is the probability that the other coin is gold? Let G be the event that the first coin is gold Pr[G ] = / Let G be the event that the second coin is gold Pr[G G ] = Pr[G G ]/ Pr[G ] = (/) / (/) = / Note: G and G are not independent Another Parado Consider a family with two children. Given that one of the children is a boy, what is the probability that both children are boys? / Tree Diagram Tree Diagram / / / / girl boy girl boy / / / / / / / girl boy girl boy / / / / boy girl boy / / / 7

8 Monty Hall Problem The host show hides a car behind one of doors at random. Behind the other two doors are goats. You select a door. Announcer opens one of others with no prize. You can decide to eep or switch. This is Tricy! We are inclined to thin: After one door is opened, others are equally liely What to do? To switch or not to switch? Monty Hall Problem Sample space = { car behind door, car behind door, car behind door } ach has probability / Staying we win if we choose the correct door Pr[ choosing correct door ] = / Switching we win if we choose the incorrect door Pr[choosing incorrect door ] = / Monty Hall Problem Let the doors be called X, Y and Z. Let C, Cy, Cz be the events that the car is behind door X and so on. Let H, Hy, Hz be the events that the host opens door X and so on. Supposing that you choose door X, the possibility that you win a car if you switch is Pr[Hy Cz] + P[Hz Cy] = Pr[Hy Cz] P[Cz] + Pr[Hz Cy] Pr[Cy] = / + / = / Bayes formula Sometimes we need to now Pr[F ], but all we now is the reverse direction Pr[ F]. Law of Total Probability Theorem. Let F, F,, F n partition of a sample space. Then Theorem. P[F ] Pr[ F] Pr[F] n Pr[ F ] n Pr[ F ] Pr[F ] Proof. Proof. Note F are mutually eclusive Pr[F ] Pr[F ] Pr[ F] Pr[ F]Pr[F] U n F 8

9 Problem A fair die is tossed and its outcome is denoted by X. After that, X independent fair coins are tossed and the number of heads obtained is denoted by Y. Compute: Pr[Y=] ercise By the law of total probability Pr[Y ] Pr[X ] is /. Compute Pr[Y= X ] Pr[Y X ] *Pr[X ] Pr[Y X ] / Use of Generating Functions Compute the probability of rolling a sum of 8 on standard dice. Problem Create a generating polynomial for the problem Compute the probability of rolling a sum of 8 on standard dice. Tae the coefficient by 8. Note, 0 0 Sample and vents Conditional Probability Bayes Law Law of Total Probability Thus, the coefficient by 8 is Here s What You Need to Know 9

### 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

### 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,

### 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

### Lecture 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

### Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

### 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

### 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

### Section 2.1. Tree Diagrams

Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.

### Probability and Random Variables (Rees: )

Probability and Random Variables (Rees:. -.) Earlier in this course, we looked at methods of describing the data in a sample. Next we would like to have models for the ways in which data can arise. Before

### The sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].

Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real

### 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 ()

### 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

### 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

### MATH 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

### 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)

### 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

### I. 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

### 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,

### 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

### 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

### Finite and discrete probability distributions

8 Finite and discrete probability distributions To understand the algorithmic aspects of number theory and algebra, and applications such as cryptography, a firm grasp of the basics of probability theory

### Introduction and Overview

Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering

### 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

### 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

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

### 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

### Chapter 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

### 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............................

### Worked 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

### 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,

### 33 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

### 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

### PROBABILITY 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

### 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.

### Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab

Lecture 2 Probability BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random

### 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

### An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### 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

### 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).

### 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

### Math 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141

Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall

### Sample 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.

### C.4 Tree Diagrams and Bayes Theorem

A26 APPENDIX C Probability and Probability Distributions C.4 Tree Diagrams and Bayes Theorem Find probabilities using tree diagrams. Find probabilities using Bayes Theorem. Tree Diagrams A type of diagram

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### 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

### 34 Probability and Counting Techniques

34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.

### Chapter 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

### 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

### Introduction to Probability

Massachusetts Institute of Technology Course Notes 0 6.04J/8.06J, Fall 0: Mathematics for Computer Science November 4 Professor Albert Meyer and Dr. Radhika Nagpal revised November 6, 00, 57 minutes Introduction

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### Possibilities and Probabilities

Possibilities and Probabilities Counting The Basic Principle of Counting: Suppose that two experiments are to be performed. Then if experiment 1 can result in any one of m possible outcomes and if, for

### Business Statistics 41000: Probability 1

Business Statistics 41000: Probability 1 Drew D. Creal University of Chicago, Booth School of Business Week 3: January 24 and 25, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:

### 15 Chances, Probabilities, and Odds

15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces

### Most of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.

PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe

### 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

### 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

### Chapter 10: Introducing Probability

Chapter 10: Introducing Probability Randomness and Probability So far, in the first half of the course, we have learned how to examine and obtain data. Now we turn to another very important aspect of Statistics

### Chapter 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.

### 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.

### 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

### A 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

### 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

### Avoiding the Consolation Prize: The Mathematics of Game Shows

Avoiding the Consolation Prize: The Mathematics of Game Shows STUART GLUCK, Ph.D. CENTER FOR TALENTED YOUTH JOHNS HOPKINS UNIVERSITY STU@JHU.EDU CARLOS RODRIGUEZ CENTER FOR TALENTED YOUTH JOHNS HOPKINS

### The Binomial Distribution

The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing

### 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

### Probability Life is full of uncertainty. Probability is the best way we currently have to quantify it.

Probability Life is full of uncertainty. Probability is the best way we currently have to quantify it. Probability Life is full of uncertainty. Probability is the best way we currently have to quantify

### Example: 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

### Probability 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

### Lesson 1: Experimental and Theoretical Probability

Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.

### Probability. 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

### 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

### 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

### 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

### 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

### Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations)

Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations) Note: At my school, there is only room for one math main lesson block in ninth grade. Therefore,

### 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

### PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

### Week 2: Conditional Probability and Bayes formula

Week 2: Conditional Probability and Bayes formula We ask the following question: suppose we know that a certain event B has occurred. How does this impact the probability of some other A. This question

### Section Tree Diagrams. Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 12.5 Tree Diagrams What You Will Learn Counting Principle Tree Diagrams 12.5-2 Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed

### Massachusetts Institute of Technology

n (i) m m (ii) n m ( (iii) n n n n (iv) m m Massachusetts Institute of Technology 6.0/6.: Probabilistic Systems Analysis (Quiz Solutions Spring 009) Question Multiple Choice Questions: CLEARLY circle the

### 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

### 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%,

### MAS108 Probability I

1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

### Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

### JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability)

JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability) Author: Daniel Cooney Reviewer: Rachel Zax Last modified: January 4, 2012 Reading from Meyer, Mathematics

### 7.1 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.

### 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.

### R Simulations: Monty Hall problem

R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem R Simulations: Monty Hall

### STAT 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,

### 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

### 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

### Using Laws of Probability. Sloan Fellows/Management of Technology Summer 2003

Using Laws of Probability Sloan Fellows/Management of Technology Summer 2003 Uncertain events Outline The laws of probability Random variables (discrete and continuous) Probability distribution Histogram

### Probability & 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

### 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.

### Chapter 5 Discrete Probability Distribution. Learning objectives

Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able

### The Monty Hall Problem

Gateway to Exploring Mathematical Sciences (GEMS) 9 November, 2013 Lecture Notes The Monty Hall Problem Mark Huber Fletcher Jones Foundation Associate Professor of Mathematics and Statistics and George

### 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