# 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:

Size: px
Start display at page:

Download "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:"

Transcription

1 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 statements such as the following: 1. There is a 30% chance of a magnitude 8 earthquake in Northern California before The average time between system failures is about three days. 3. The chance of getting a flush in a five-card poker hand is about 2 in In this load-balancing scheme, the probability that any processor has to deal with more than twelve requests for service is negligible. Implicit in all such statements is the notion of an underlying probability space. This may be the result of some model we build of the real world (as in 1 and 2 above), or of a random experiment that we have ourselves constructed (as in 3 and 4 above). None of these statements makes sense unless we specify the probability space we are talking about: for this reason, statements like 1 (which are typically made without this context) are almost content-free. Probability spaces Every probability space is based on a random experiment, such as rolling a die, shuffling a deck of cards, picking a number, assigning jobs to processors, running a system etc. Rather than attempt to define experiment directly, we shall define it by its set of possible outcomes, which we call a sample space. Note that all outcomes must be disjoint, and they must cover all possibilities. Definition 17.1 (sample space): The sample space of an experiment is the set of all possible outcomes. An outcome is often called a sample point or atomic event. Definition 17.2 (probability space): A probability space is a sample space Ω, together with a probability Pr[ω] for each sample point ω, such that 0 Pr[ω] 1 for all ω Ω. ω Ω Pr[ω] = 1, i.e., the sum of the probabilities of all outcomes is 1. [Strictly speaking, what we have defined above is a restricted set of probability spaces known as discrete spaces: this means that the set of sample points is either finite or countably infinite (such as the natural numbers, or the integers, or the rationals, but not the real numbers). Later, we will talk a little about continuous sample spaces, but for now we assume everything is discrete.] CS 70, Spring 2006, Lecture 17 1

2 Here are some examples of (discrete) probability spaces: 1. Flip a fair coin. Here Ω = {H,T}, and Pr[H] = Pr[T] = Flip a fair coin three times. Here Ω = {(t 1,t 2,t 3 ) : t i {H,T }}, where t i gives the outcome of the ith toss. Thus Ω consists of 2 3 = 8 points, each with equal probability 1 8. More generally, if we flip the coin n times, we get a sample space of size 2 n (corresponding to all words of length n over the alphabet {H,T }), each point having probability 1 2. n 3. Flip a biased coin three times. Suppose the bias is two-to-one in favor of Heads, i.e., it comes up Heads with probability 2 3 and Tails with probability 1 3. The sample space here is exactly the same as in the previous example. However, the probabilities are different. For example, Pr[HHH] = = 8 27, while Pr[T HH] = = [Note: We have cheerfully multiplied probabilities here; we ll explain why this is OK later. It is not always OK!] More generally, if we flip a biased coin with Heads probability p (and Tails probability 1 p) n times, the probability of a given sequence is p r (1 p) n r, where r is the number of H s in the sequence. Biased coin-tossing sequences show up in many contexts: for example, they might model the behavior of n trials of a faulty system, which fails each time with probability p. 4. Roll two dice. Then Ω = {(i, j) : 1 i, j 6}. Each of the 36 outcomes has equal probability, Card Shuffling. Shuffle a deck of cards. Here Ω consists of the permutations of the deck, each with equal probability 1. [Note that we re really talking about an idealized mathematical model of shuffling here; in real life, there will always be a bit of bias in our shuffling. However, the mathematical model is close enough to be useful.] 6. Poker Hands. Shuffle a deck of cards, and then deal a poker hand. Here Ω consists of all possible five-card hands, each with equal probability (because the deck is assumed to be randomly shuffled). The number of such hands is ( 52) 5, i.e., the number of ways of choosing five cards from the deck of 52 (without worrying about the order). As we saw many lectures ago, ( 52) 5 = 5!47! = = Balls and Bins. Throw 20 balls into 10 bins, so that each ball is equally likely to land in any bin, regardless of what happens to the other balls. Here Ω = {(b 1,b 2,...,b 20 ) : 1 b i 10}; the component b i denotes the bin in which ball i lands. There are possible outcomes (why?), each with probability 1. More generally, if we throw m balls into n bins, we have a sample space of size n m [Note that example 2 above is a special case of balls and bins, with m = 3 and n = 2.] As we shall see, balls and bins is another probability space that shows up very often in Computer Science: for example, we can think of it as modeling a load balancing scheme, in which each job is sent to a random processor. 8. The Monty Hall Problem. In an (in)famous 1970s game show hosted by one Monty Hall, a contestant was shown three doors; behind one of the doors was a prize, and behind the other two were goats. The contestant picks a door (but doesn t open it), and Hall s assistant (Carol) opens one of the other two doors, revealing a goat. The contestant is then given the option of sticking with his current door, or switching to the other unopened one. He wins the prize if and only if his chosen door is the correct one. The question, of course, is: Does the contestant have a better chance of winning if he switches doors? What is the sample space here? Well, we can describe the outcome of the game (up to the point where the contestant makes his final decision) using a triple of the form (i, j,k), where i, j,k {1,2,3}. The CS 70, Spring 2006, Lecture 17 2

3 Events values i, j, k respectively specify the location of the prize, the initial door chosen by the contestant, and the door opened by Carol. Note that some triples are not possible: e.g., (1,2,1) is not, because Carol never opens the prize door. Thinking of the sample space as a tree structure, in which first i is chosen, then j, and finally k (depending on i and j), we see that there are exactly 12 sample points. Assigning probabilities to the sample points here requires pinning down some assumptions: The prize is equally likely to be behind any of the three doors. Initially, the contestant is equally likely to pick any of the three doors. If the contestant happens to pick the prize door (so there are two possible doors for Carol to open), Carol is equally likely to pick either one. From this, we can assign a probability to every sample point. For example, the point (1, 2, 3) corresponds to the prize being placed behind door 1 (with probability 1 3 ), the contestant picking door 2 (with probability 1 3 ), and Carol opening door 3 (with probability 1, because she has no choice). So Pr[(1,2,3)] = = 1 9. [Note: Again we are multiplying probabilities here, without proper justification!] Note that there are six outcomes of this type, characterized by having i j (and hence k must be different from both). On the other hand, we have Pr[(1,1,2)] = = And there are six outcomes of this type, having i = j. These are the only possible outcomes, so we have completely defined our probability space. Just to check our arithmetic, we note that the sum of the probabilities of all outcomes is (6 1 9 )+( ) = 1. In the Monty Hall problem, we are interested in the probability that the constestant wins the prize. This is itself not a single outcome (the contestant can win in several different ways), but a set of outcomes. This leads us to: Definition 17.3 (event): An event A in a sample space Ω is any subset A Ω. How should we define the probability of an event A? Naturally, we should just add up the probabilities of the sample points in A. Definition 17.4 (probability of an event): For any event A Ω, we define the probability of A to be Pr[A] = Pr[ω]. ω A Let s look at some examples; the number of the example refers to the probability space in our previous list. 1. Fair coin. Let A be the event the coin comes up Heads. Then Pr[A] = Three fair coins. Let A be the event that all three coin tosses are the same. Then Pr[A] = Pr[HHH]+ Pr[T T T] = = Biased coins. Let A be the same event as in the previous example. Then Pr[A] = Pr[HHH]+ Pr[T T T] = = 9 27 = 1 3. As a second example, let B be the event that there are exactly two CS 70, Spring 2006, Lecture 17 3

4 Heads. We know that the probability of any outcome with two Heads (and therefore one Tail) is ( 2 3 )2 ( 1 3 ) = How many such outcomes are there? Well, there are ( 3 2) = 3 ways of choosing the positions of the Heads, and these choices completely specify the sequence. So Pr[B] = = 4 9. More generally, the probability of getting exactly r Heads from n tosses of a biased coin with Heads probability p is ( n r) p r (1 p) n r. 4. Dice. Let A be the event that the sum of the dice is at least 10, and B the event that there is at least one 6. Then Pr[A] = 6 36 = , and Pr[B] = 36. In this example (and in 1 and 2 above), our probability space is uniform, i.e., all the sample points have the same probability (which must be 1 Ω, where Ω denotes the size of Ω). In such circumstances, the probability of any event A is clearly just Pr[A] = # of sample points in A # of sample points in Ω = A Ω. So for uniform spaces, computing probabilities reduces to counting sample points! 6. Card shuffling. Let A be the event that the top card is an ace. Then by the above remarks Pr[A] = # of permutations with an ace on top. How many permutations have an ace on top? Well, there are four choices for the ace; and once we have chosen it and put it on top, there are exactly 51! ways to arrange the remaining 51 cards. So the number of such permutations is 4 51!. Thus Pr[A] = 4 51! = 4 52 = Poker hands. What is the probability that our poker hand is a flush? [For those who are not addicted to gambling, a flush is a hand in which all cards have the same suit, say Hearts.] To compute this probability, we just need to figure out how many poker hands are flushes. Well, there are 13 cards in each suit, so the number of flushes in each suit is ( 13) 5. The total number of flushes is therefore 4 ( 13 ) 5. So we have Pr[hand is a flush] = 4 (13 ) 5 ) = ( ! 5! 47! 5! 8! = Balls and bins. Let A be the event that bin 1 is empty. Again, we just need to count how many outcomes have this property. And this is exactly the number of ways all 20 balls can fall into the remaining nine boxes, which is Hence Pr[A] = 920 = ( ) What is the probability that bin 1 contains at least one ball? This is easy: this event, call it Ā, is the complement of A, i.e., it consists of precisely those sample points that are not in A. So Pr[Ā] = 1 Pr[A] More generally, if we throw m balls into n bins, we have ( ) n 1 m ( Pr[bin 1 is empty] = = 1 1 m. n n) 9. Monty Hall. Let s return to the Monty Hall problem. Recall that we want to investigate the relative merits of the sticking strategy and the switching strategy. Let s suppose the contestant decides to switch doors. The event A we are interested in is the event that the contestant wins. Which sample points (i, j,k) are in A? Well, since the contestant is switching doors, his initial choice j cannot be equal to the prize door, which is i. And all outcomes of this type correspond to a win for the contestant, because Carol must open the second non-prize door, leaving the contestant to switch to the prize door. So A consists of all outcomes of the first type in our earlier analysis; recall that there are six of these, CS 70, Spring 2006, Lecture 17 4

5 each with probability 1 9. So Pr[A] = 6 9 = 2 3. I.e., using the switching strategy, the contestant wins with probability 2 3! It should be intuitively clear (and easy to check formally try it!) that under the sticking strategy his probability of winning is 1 3. (In this case, he is really just picking a single random door.) So by switching, the contestant actually improves his odds by a huge amount! This is one of many examples that illustrate the importance of doing probability calculations systematically, rather than intuitively. Recall the key steps in all our calculations: What is the sample space (i.e., the experiment and its set of possible outcomes)? What is the probability of each outcome (sample point)? What is the event we are interested in (i.e., which subset of the sample space)? Finally, compute the probability of the event by adding up the probabilities of the sample points inside it. Whenever you meet a probability problem, you should always go back to these basics to avoid potential pitfalls. Even experienced researchers make mistakes when they forget to do this witness many erroneous proofs, submitted by mathematicians to newspapers at the time, of the fact that the switching strategy in the Monty Hall problem does not improve the odds. CS 70, Spring 2006, Lecture 17 5

### 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

### 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

### 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

### 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

### 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

### 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

### Theory of Probability - Brett Bernstein

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

### 23 Applications of Probability to Combinatorics

November 17, 2017 23 Applications of Probability to Combinatorics William T. Trotter trotter@math.gatech.edu Foreword Disclaimer Many of our examples will deal with games of chance and the notion of gambling.

### Probability. Dr. Zhang Fordham Univ.

Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!

### Section Summary. Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning

Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event Pierre-Simon Laplace (1749-1827) We first study Pierre-Simon

### Counting and Probability

Counting and Probability Lecture 42 Section 9.1 Robb T. Koether Hampden-Sydney College Wed, Apr 9, 2014 Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 1 / 17 1 Probability

### Junior Circle Meeting 5 Probability. May 2, ii. In an actual experiment, can one get a different number of heads when flipping a coin 100 times?

Junior Circle Meeting 5 Probability May 2, 2010 1. We have a standard coin with one side that we call heads (H) and one side that we call tails (T). a. Let s say that we flip this coin 100 times. i. How

### 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

### 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.

### 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

### 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

### 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

### Introductory Probability

Introductory Probability Combinations Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Assigning Objects to Identical Positions Denitions Committee Card Hands Coin Toss Counts

### 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

### CS 361: Probability & Statistics

January 31, 2018 CS 361: Probability & Statistics Probability Probability theory Probability Reasoning about uncertain situations with formal models Allows us to compute probabilities Experiments will

### 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. 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

### 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

### 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

### 1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,

### 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

### 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.

### 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

### Finite Mathematics MAT 141: Chapter 8 Notes

Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication

### 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

### 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

### 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

Probability Paradoxes Washington University Math Circle February 20, 2011 1 Introduction We re all familiar with the idea of probability, even if we haven t studied it. That is what makes probability so

### MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG

MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices

### 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.

### If a regular six-sided die is rolled, the possible outcomes can be listed as {1, 2, 3, 4, 5, 6} there are 6 outcomes.

Section 11.1: The Counting Principle 1. Combinatorics is the study of counting the different outcomes of some task. For example If a coin is flipped, the side facing upward will be a head or a tail the

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

### Statistics Intermediate Probability

Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting

### 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

### 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

### Problems for Recitation 17

6.042/18.062J Mathematics for Computer Science November 10, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 17 The Four-Step Method This is a good approach to questions of the form, What

### 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.

### COUNTING AND PROBABILITY

CHAPTER 9 COUNTING AND PROBABILITY It s as easy as 1 2 3. That s the saying. And in certain ways, counting is easy. But other aspects of counting aren t so simple. Have you ever agreed to meet a friend

### 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

### Before giving a formal definition of probability, we explain some terms related to probability.

probability 22 INTRODUCTION In our day-to-day life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely

### 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

### Name. Is the game fair or not? Prove your answer with math. If the game is fair, play it 36 times and record the results.

Homework 5.1C You must complete table. Use math to decide if the game is fair or not. If Period the game is not fair, change the point system to make it fair. Game 1 Circle one: Fair or Not 2 six sided

### 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

### The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you \$5 that if you give me \$10, I ll give you \$20.)

The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you \$ that if you give me \$, I ll give you \$2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If

### CS 361: Probability & Statistics

February 7, 2018 CS 361: Probability & Statistics Independence & conditional probability Recall the definition for independence So we can suppose events are independent and compute probabilities Or we

### 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.

### 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

### Section Introduction to Sets

Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase

### 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,

### 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

### CS1802 Week 9: Probability, Expectation, Entropy

CS02 Discrete Structures Recitation Fall 207 October 30 - November 3, 207 CS02 Week 9: Probability, Expectation, Entropy Simple Probabilities i. What is the probability that if a die is rolled five times,

### 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

### STAT 155 Introductory Statistics. Lecture 11: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 11: Randomness and Probability Model 10/5/06 Lecture 11 1 The Monty Hall Problem Let s Make A Deal: a game show

### Probability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability

Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write

### Probability and the Monty Hall Problem Rong Huang January 10, 2016

Probability and the Monty Hall Problem Rong Huang January 10, 2016 Warm-up: There is a sequence of number: 1, 2, 4, 8, 16, 32, 64, How does this sequence work? How do you get the next number from the previous

### 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

### 1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.

1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find

### 4.3 Rules of Probability

4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:

### 1. How to identify the sample space of a probability experiment and how to identify simple events

Statistics Chapter 3 Name: 3.1 Basic Concepts of Probability Learning objectives: 1. How to identify the sample space of a probability experiment and how to identify simple events 2. How to use the Fundamental

### Combinatorics and Intuitive Probability

Chapter Combinatorics and Intuitive Probability The simplest probabilistic scenario is perhaps one where the set of possible outcomes is finite and these outcomes are all equally likely. A subset of the

### Introduction to Probability

6.04/8.06J Mathematics for omputer Science Srini Devadas and Eric Lehman pril 4, 005 Lecture Notes Introduction to Probability Probability is the last topic in this course and perhaps the most important.

### 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

### Elementary Statistics. Basic Probability & Odds

Basic Probability & Odds What is a Probability? Probability is a branch of mathematics that deals with calculating the likelihood of a given event to happen or not, which is expressed as a number between

### Lecture 2: Sum rule, partition method, difference method, bijection method, product rules

Lecture 2: Sum rule, partition method, difference method, bijection method, product rules References: Relevant parts of chapter 15 of the Math for CS book. Discrete Structures II (Summer 2018) Rutgers

### 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

### 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

### Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37

Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete

### 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,

### 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

### 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

### STOR 155 Introductory Statistics. Lecture 10: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model 10/6/09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal: a game show

### Grade 7/8 Math Circles February 25/26, Probability

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely

### CS1800: Intro to Probability. Professor Kevin Gold

CS1800: Intro to Probability Professor Kevin Gold Probability Deals Rationally With an Uncertain World Using probabilities is the only rational way to deal with uncertainty De Finetti: If you disagree,

### CSC/MATA67 Tutorial, Week 12

CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly

### 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

### 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

### Random Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment.

Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

### 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

### CS 787: Advanced Algorithms Homework 1

CS 787: Advanced Algorithms Homework 1 Out: 02/08/13 Due: 03/01/13 Guidelines This homework consists of a few exercises followed by some problems. The exercises are meant for your practice only, and do

### Chapter 5 - Elementary Probability Theory

Chapter 5 - Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling

### Probability Theory. POLI Mathematical and Statistical Foundations. Sebastian M. Saiegh

POLI 270 - Mathematical and Statistical Foundations Department of Political Science University California, San Diego November 11, 2010 Introduction to 1 Probability Some Background 2 3 Conditional and

### 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

### 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

### Random Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }

Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Multiple Events and Combined Probabilities 1 Determine the probability of each of the following events assuming that the die has equal probability

### 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

### Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples

Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 6.1 An Introduction to Discrete Probability Page references correspond to locations of Extra Examples icons in the textbook.

### RANDOM EXPERIMENTS AND EVENTS

Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting

### 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

### Key Concepts. Theoretical Probability. Terminology. Lesson 11-1

Key Concepts Theoretical Probability Lesson - Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally

### Bellwork Write each fraction as a percent Evaluate P P C C 6

Bellwork 2-19-15 Write each fraction as a percent. 1. 2. 3. 4. Evaluate. 5. 6 P 3 6. 5 P 2 7. 7 C 4 8. 8 C 6 1 Objectives Find the theoretical probability of an event. Find the experimental probability

### 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,