# Theory of Probability - Brett Bernstein

Size: px
Start display at page:

Transcription

1 Theory of Probability - Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of getting at least 1 head.. You are flipping a coin twice that has.75 chance of landing heads. What is the probability of getting 1 head? 3. Let S {1,,..., 100} with all outcomes equally likely. What is the probability of the event that a chosen integer isn t divisible by, 3, or 5? 4. We want to distribute 10 identical pieces of candy to 4 (distinguishable children. For each piece of candy we roll a 4-sided die and give it to the associated child. What is the probability that the children receive, 5, 1, pieces of candy, respectively (i.e., first child receives, second receives 5, etc.? 5. Suppose you are dealt an ordered sequence of k cards from a 5 card deck without replacement. Compute the probability of getting a particular k card (unordered hand. Solutions 1. Let S {(a 1, a : a i is H or T } with all outcomes equally likely. This intuitive choice for the probability measure will later be seen to come from the fact that the coin flips are modeled as independent of each other. Then we have P (at least one head P ({HT, T H, HH} 3 4, where we wrote (H, T as HT for brevity.. Let S {(a 1, a : a i {H, T }}. We define P by P ({(a 1, a }.75 # of heads # of tails.5 using the general finite space. As above, we will see later that this intuitive choice of measure comes from modeling the coin flips as independent. Note that The result is (.75( (.75(

2 3. Let D, D 3, D 5 denote the subsets of S divisible by, 3, 5 respectively. Then D 50, D 3 33, D 5 0, D D 3 16, D D 5 10, D 3 D 5 6, D D 3 D 5 3. This gives The answer is D D 3 D Here S {(d 1,..., d 10 : d i {1,, 3, 4}} and all outcomes are equally likely. The probability is thus ( 10,5,1, 4 10 since S 4 10 and there are ( 10 10!, 5, 1,!5!! elements of S that give the correct counts. Thus an un Each ordered hand occurs with an equal probability of ordered hand occurs with probability 5 51 (5 k+1. k! 5 51 (5 k + 1 ( 1 5 k as expected. Aside on Measure Theory Example 1 (Infinite Coin Flipping. Suppose we have a fair coin and wish to flip it infinitely many times. We could use the following sample space: S {(a 1, a,... : a i {H, T }}. If the model accurately reflects coin flipping, we would want the following calculations to hold: 1. The probability the first flip is heads should be 1/.. The probability the first two flips are heads should be 1/4. 3. In general, if we specify the first k flips, we expect the probability to be 1/ k.

3 This leads to the uncomfortable situation that the probability of any particular sequence is 0 since it must be less than 1/ k for all k by monotonicity. Thus any measure we choose must assign positive probability to some events, while assigning zero probability to every singleton event. Formulaically, we must have P ({s} 0, for any s S. One key aspect to why this is possible is that S is an uncountable set, so it can have probability 1 even though all the singletons have probability zero. This is one of the nuances of countable additivity. Example (Aside: Lebesgue Measure. Let S [0, 1] and define P ( (a, b b a. For more general events A we define { } P (A inf (b i a i : (a i, b i A. That is, we approximate general sets by looking at coverings by intervals. The infimum forces us to look at tighter and tighter coverings. In addition to allowing us to deal with sets of real numbers, we can use this model to study sequences of fair coin flips by treating each sequence as a binary expansion. One odd consequence of this theory is that not every subset of [0, 1] can be called an event if we want countable additivity to hold. As such, we restrict to a special class of events called the measurable sets. A Few More Interesting Examples 1. Assuming you are randomly dealt 5 cards from a standard deck of playing cards, what is the probability of getting two pair (a poker hand? What about a full house?. How many students are needed in a class before it is likely that at least have the same birthday (assume 365 days in a year, each day equally likely? 3. Suppose n people put their ID cards down, the IDs are shuffled, and then randomly returned. What is the probability nobody gets their ID back? What if n is very large? Solutions 1. Consider the sample space of all 5 card subsets of the 5 cards, where each is equally likely. To count the number of possible ways to get two pair, we first choose the two values for the pairs, and the value of the remaining card. Then we choose the suits for each pair and the extra card. ( 13 ( 11 4 ( 4 (

4 The probability of a full house is 13 1 (4 4 3( ( We could also count these using ordered hands. For two pair we have !!!! Here we count all possible ways of choosing hands in the order 113, where the numbers denote values. Then we must consider all orderings of this string, but divide by an extra! since swapping the 1 s and s doesn t yield a distinct hand. To me this calculation seems trickier. For full house we have ! 3!! students. The probability of k students and no matches is (365 k k. Here the sample space is the collection of all sequences of length k where each value is between 1 and 365, and each sequence is equally likely. Even though 3 seems small compared to 365, the number of pairs of people is actually ( The probability is roughly 1/e. To compute the precise probability we use inclusionexclusion. Our sample space is the collection of all permutations of 1,..., n with each equally likely (there are n! of them. Let A i denote the event that the ith person picks up his ID (i.e., that the permutation has i in the ith spot. Then we can compute the probability of at least one person getting their correct ID by inclusion-exclusion: P Note that ( n A i n P (A i i<j P (A i A j + + ( 1 n 1 P (A 1 A n. P (A i 1/n, P (A i A j 1/(n(n 1, P (A i A j A k 1/(n(n 1(n,.... In general, if there are p distinct A i events we are intersecting the probability is (n p! n!. 4

5 This gives ( n P A i n 1 ( n n 1 n(n ( 1n 1 n! n ( n (n k! ( 1 k k n! k1 n ( 1 k 1 k! k1 1 e 1, where the approximation improves as n grows (error is less than 1/n!. The answer to our problem is 1/e. It is interesting that the probability converges to a number strictly between 0 and 1. One intuition for the result is that the events are weakly dependent and well approximated by independent events. Conditional Probability Conditioning Formula Conditioning is the process where we update our beliefs (probabilities in light of newly found information. As you can imagine, this is a very practical skill to master. It turns out that conditioning can also be used to greatly simplify a calculation, by using the Law of Total Probability. Joseph Blitzstein says Conditioning is the Soul of Statistics. Let A, B be events with P (B > 0. We use the notation P (A B (read probability of A given B denote the probability that A assuming B will occur. In other words, assuming we have the information that B will occur, P (A B is the probability that A will occur. Formally, it is defined to be P (A B P (A B. P (B We also write P (A B, C P (A B C P (A BC. The motivation for this formula can be understood by looking at examples of sample spaces, and drawing Venn diagrams. It can also be instructive to look at the formula in the following form: P (A B P (A BP (B. That is, the probability of A and B both occurring is the probability of B occurring, times the probability of A occurring given that B has occurred. This can be repeated iteratively, to get the following formula: Theorem 3. Assuming P (E 1 > 0,..., P (E n > 0, P (E 1 E E n P (E 1 P (E E 1 P (E 3 E 1, E P (E n E 1,..., E n 1. 5

6 Proof. We simply write out all the formulas and cancel: P (E 1 P (E E 1 P (E 3 E E 1 P (E n E 1 P (E 1 P (E E 1 P (E n 1 E 1 P (E n E 1. Example 4 (Tree Diagrams. We are going to make 3 coin flips, but with the following rules. Each time you get a heads, the next coin should have half the chance of getting heads (i.e., if your first flip is heads, the second coin will have 1/4 probability of heads, and 3/4 probability of tails. What are the probabilities of getting 0,1,,3 heads? One way to solve this problem is to draw a tree whose leaves are all 8 possibilities. The probabilities on each edge are conditional, and to compute the probabilities of the leaves, we are using the previous theorem. Conditioning Exercises 1. A jar contains a 4-sided die and a 6-sided die. We uniformly at random pick a die out of the jar and roll it. Assuming the roll is 3, what is the chance we picked the 6-sided die? Do you expect it to be bigger or smaller than 1/?. Flip a fair coin 5 times in a row. Assuming we get a total of 3 heads, what is the probability the first flip was heads? 3. A standard deck is shuffled, and two cards are dealt face down. The dealer looks at both cards. Solutions (a If the dealer tells you At least one of the cards is an Ace, what is the probability there are two Aces? (b If the dealer tells you One of the cards is the Ace of Spades, what is the probability there are two Aces? 1. Let A be the event of rolling a 3, and B be the event of drawing the 6-sided die. Then we have Then we have P (A B 1/6 1 P (B A P (A B P (A and P (A /1 1/1 + 1/ / 5. Note that we would have gotten the same result for rolls 1,, 3, 4. For roll 5, 6 we know it is 6-sided with probability 1. 6

7 . Let A be the event of getting 3 heads, and let B be the event that the first flip is heads. Then ( ( P (A and P (A B Thus we have P (B A P (A B P (A ( 4 ( (a Let A denote the event of getting two aces, and B the event of getting at least one ace. Then we have Thus the answer is P (B P (A B P (B and P (A B (b Let A denote the event of getting two aces, and B the event of getting the ace of spaces. Then we have Thus the answer is P (B P (A B P (B and P (A B

### Counting and Probability Math 2320

Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A

### The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:

CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such

### The topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:

CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of

### Intermediate Math Circles November 1, 2017 Probability I

Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.

### Chapter 2. Permutations and Combinations

2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find

### Chapter 1. Probability

Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

### Compound Probability. Set Theory. Basic Definitions

Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic

### Discrete Structures Lecture Permutations and Combinations

Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these

### Block 1 - Sets and Basic Combinatorics. Main Topics in Block 1:

Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.

### Probability. Engr. Jeffrey T. Dellosa.

Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional

### Chapter 1. Probability

Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

### 4.1 Sample Spaces and Events

4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an

### The study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability

The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch

### 3. Discrete Probability. CSE 312 Spring 2015 W.L. Ruzzo

3. Discrete Probability CSE 312 Spring 2015 W.L. Ruzzo 2 Probability theory: an aberration of the intellect and ignorance coined into science John Stuart Mill 3 sample spaces Sample space: S is a set of

### 2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)

2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is

### Important Distributions 7/17/2006

Important Distributions 7/17/2006 Discrete Uniform Distribution All outcomes of an experiment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then

### Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1

### November 6, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern

### Section : Combinations and Permutations

Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 6 Lecture Notes Discrete Probability Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. Introduction and

### Probability Rules. 2) The probability, P, of any event ranges from which of the following?

Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,

### Discrete Structures for Computer Science

Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is

### A Probability Work Sheet

A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we

### 8.2 Union, Intersection, and Complement of Events; Odds

8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context

### Such a description is the basis for a probability model. Here is the basic vocabulary we use.

5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these

### Probability. Ms. Weinstein Probability & Statistics

Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random

### CHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events

CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes

### The probability set-up

CHAPTER 2 The probability set-up 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample

### 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

### CSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions)

CSE 31: Foundations of Computing II Quiz Section #: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions) Review: Main Theorems and Concepts Binomial Theorem: x, y R, n N: (x + y) n

### EECS 203 Spring 2016 Lecture 15 Page 1 of 6

EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including

### Problem Set 2. Counting

Problem Set 2. Counting 1. (Blitzstein: 1, Q3 Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his favorite ten restaurants. i

### Foundations of Computing Discrete Mathematics Solutions to exercises for week 12

Foundations of Computing Discrete Mathematics Solutions to exercises for week 12 Agata Murawska (agmu@itu.dk) November 13, 2013 Exercise (6.1.2). A multiple-choice test contains 10 questions. There are

### CSCI 2200 Foundations of Computer Science (FoCS) Solutions for Homework 7

CSCI 00 Foundations of Computer Science (FoCS) Solutions for Homework 7 Homework Problems. [0 POINTS] Problem.4(e)-(f) [or F7 Problem.7(e)-(f)]: In each case, count. (e) The number of orders in which a

### Lecture 14. What s to come? Probability. A bag contains:

Lecture 14 What s to come? Probability. A bag contains: What is the chance that a ball taken from the bag is blue? Count blue. Count total. Divide. Today: Counting! Later: Probability. Professor Walrand.

### 1. The chance of getting a flush in a 5-card poker hand is about 2 in 1000.

CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 15 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette wheels. Today

### MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology

MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally

### The Product Rule can be viewed as counting the number of elements in the Cartesian product of the finite sets

Chapter 6 - Counting 6.1 - The Basics of Counting Theorem 1 (The Product Rule). If every task in a set of k tasks must be done, where the first task can be done in n 1 ways, the second in n 2 ways, and

### Chapter 3: Elements of Chance: Probability Methods

Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,

### Probability (Devore Chapter Two)

Probability (Devore Chapter Two) 1016-351-01 Probability Winter 2011-2012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................

### Probability Theory. Mohamed I. Riffi. Islamic University of Gaza

Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)

### STAT 430/510 Probability Lecture 1: Counting-1

STAT 430/510 Probability Lecture 1: Counting-1 Pengyuan (Penelope) Wang May 22, 2011 Introduction In the early days, probability was associated with games of chance, such as gambling. Probability is describing

### Week 3 Classical Probability, Part I

Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability

### CIS 2033 Lecture 6, Spring 2017

CIS 2033 Lecture 6, Spring 2017 Instructor: David Dobor February 2, 2017 In this lecture, we introduce the basic principle of counting, use it to count subsets, permutations, combinations, and partitions,

### The probability set-up

CHAPTER The probability set-up.1. Introduction and basic theory We will have a sample space, denoted S sometimes Ω that consists of all possible outcomes. For example, if we roll two dice, the sample space

### Probability: Terminology and Examples Spring January 1, / 22

Probability: Terminology and Examples 18.05 Spring 2014 January 1, 2017 1 / 22 Board Question Deck of 52 cards 13 ranks: 2, 3,..., 9, 10, J, Q, K, A 4 suits:,,,, Poker hands Consists of 5 cards A one-pair

### Week 1: Probability models and counting

Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model

### 1. Counting. 2. Tree 3. Rules of Counting 4. Sample with/without replacement where order does/doesn t matter.

Lecture 4 Outline: basics What s to come? Probability A bag contains: What is the chance that a ball taken from the bag is blue? Count blue Count total Divide Today: Counting! Later: Probability Professor

### November 11, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.

### WEEK 7 REVIEW. Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.1)

WEEK 7 REVIEW Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.) Definition of Probability (7.2) WEEK 8-7.3, 7.4 and Test Review THE MULTIPLICATION

### Basic Probability Models. Ping-Shou Zhong

asic Probability Models Ping-Shou Zhong 1 Deterministic model n experiment that results in the same outcome for a given set of conditions Examples: law of gravity 2 Probabilistic model The outcome of the

### n! = n(n 1)(n 2) 3 2 1

A Counting A.1 First principles If the sample space Ω is finite and the outomes are equally likely, then the probability measure is given by P(E) = E / Ω where E denotes the number of outcomes in the event

### Probability and Counting Techniques

Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each

### Axiomatic Probability

Axiomatic Probability The objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance thtat A will occur.

### STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes

STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle

### The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n

Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product

### 7.1 Experiments, Sample Spaces, and Events

7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment

### MATHEMATICS E-102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms)

MATHEMATICS E-102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms) Last modified: September 19, 2005 Reference: EP(Elementary Probability, by Stirzaker), Chapter

### Applied Statistics I

Applied Statistics I Liang Zhang Department of Mathematics, University of Utah June 12, 2008 Liang Zhang (UofU) Applied Statistics I June 12, 2008 1 / 29 In Probability, our main focus is to determine

### Mathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015

1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:

### CISC-102 Fall 2017 Week 8

Week 8 Page! of! 34 Playing cards. CISC-02 Fall 207 Week 8 Some of the following examples make use of the standard 52 deck of playing cards as shown below. There are 4 suits (clubs, spades, hearts, diamonds)

### Honors Precalculus Chapter 9 Summary Basic Combinatorics

Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each

### 18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY

18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY 1. Three closed boxes lie on a table. One box (you don t know which) contains a \$1000 bill. The others are empty. After paying an entry fee, you play the following

### Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13

CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13 Introduction to Discrete Probability In the last note we considered the probabilistic experiment where we flipped a

### Math 1313 Section 6.2 Definition of Probability

Math 1313 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

### Board Question 1. There are 5 Competitors in 100m final. How many ways can gold silver and bronze be awarded? May 27, / 28

Board Question 1 There are 5 Competitors in 100m final. How many ways can gold silver and bronze be awarded? Photograph of Usain Bolt running a race removed due to copyright restrictions. May 27, 2014

### CHAPTER 8 Additional Probability Topics

CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information

### 7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count

7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count Probability deals with predicting the outcome of future experiments in a quantitative way. The experiments

### MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability)

MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability) Last modified: November 10, 2004 This follows very closely Apostol, Chapter 13, the course pack. Attachments

### Probability is the likelihood that an event will occur.

Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main

### November 8, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol

CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 14 : Counting In this reading we discuss counting. Often, we are interested in the cardinality

### PROBABILITY. 1. Introduction. Candidates should able to:

PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation

### Lecture 1. Permutations and combinations, Pascal s triangle, learning to count

18.440: Lecture 1 Permutations and combinations, Pascal s triangle, learning to count Scott Sheffield MIT 1 Outline Remark, just for fun Permutations Counting tricks Binomial coefficients Problems 2 Outline

### CS 237: Probability in Computing

CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 5: o Independence reviewed; Bayes' Rule o Counting principles and combinatorics; o Counting considered

### Probability and Randomness. Day 1

Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of

### CSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability

CSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability Review: Main Theorems and Concepts Binomial Theorem: Principle of Inclusion-Exclusion

### Chapter 1. Set Theory

Chapter 1 Set Theory 1 Section 1.1: Types of Sets and Set Notation Set: A collection or group of distinguishable objects. Ex. set of books, the letters of the alphabet, the set of whole numbers. You can

### Chapter 4: Introduction to Probability

MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below

### Chapter 1: Sets and Probability

Chapter 1: Sets and Probability Section 1.3-1.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping

### Sample Spaces, Events, Probability

Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.

### CHAPTER 7 Probability

CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can

### What is counting? (how many ways of doing things) how many possible ways to choose 4 people from 10?

Chapter 5. Counting 5.1 The Basic of Counting What is counting? (how many ways of doing things) combinations: how many possible ways to choose 4 people from 10? how many license plates that start with

### Dependence. Math Circle. October 15, 2016

Dependence Math Circle October 15, 2016 1 Warm up games 1. Flip a coin and take it if the side of coin facing the table is a head. Otherwise, you will need to pay one. Will you play the game? Why? 2. If

### ECON 214 Elements of Statistics for Economists

ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing

### CSC/MTH 231 Discrete Structures II Spring, Homework 5

CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the

### Section The Multiplication Principle and Permutations

Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different

### Permutations and Combinations. MATH 107: Finite Mathematics University of Louisville. March 3, 2014

Permutations and Combinations MATH 107: Finite Mathematics University of Louisville March 3, 2014 Multiplicative review Non-replacement counting questions 2 / 15 Building strings without repetition A familiar

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even

### Raise your hand if you rode a bus within the past month. Record the number of raised hands.

166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

### CS100: DISCRETE STRUCTURES. Lecture 8 Counting - CH6

CS100: DISCRETE STRUCTURES Lecture 8 Counting - CH6 Lecture Overview 2 6.1 The Basics of Counting: THE PRODUCT RULE THE SUM RULE THE SUBTRACTION RULE THE DIVISION RULE 6.2 The Pigeonhole Principle. 6.3

### (a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?

Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent

### Section Summary. Permutations Combinations Combinatorial Proofs

Section 6.3 Section Summary Permutations Combinations Combinatorial Proofs Permutations Definition: A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement

### Business Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal

Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,

### Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ.

Counting (Enumerative Combinatorics) X. Zhang, Fordham Univ. 1 Chance of winning?! What s the chances of winning New York Megamillion Jackpot!! just pick 5 numbers from 1 to 56, plus a mega ball number