# Week 1: Probability models and counting

Size: px
Start display at page:

Transcription

1 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 we need the following ingredients A sample space S which is the collection of all possible outcomes of the (random experiment. We shall consider mostly finite sample space S. A probability distribution. To each element i S we assign a probability p(i S. We have p(i = Probability that the outcome i occurs 0 p(i 1 p(i = 1. By definition probabilities are nonnegative numbers and add up to 1. An event A is a subset of the sample space S. It describes an experiment or an observation that is compatible with the outcomes i A. The probability that A occurs, P (A, is given by P (A = p(i. i A i S Example: Thoss three (fair coins and record if the coin lands on tail (T or head (H. The sample space is S = {HHH, HHT, HT H, T HH, HT T, T HT, T T H, T T T } and has 8 = 2 3 elements. For fair coins it is natural assign the probability 1/8 to each outcome. P (HHH = P (HHT = = P (T T T = 1/8 An example of an event A is that at least two of the coins land on head. Then A = {HHH, HHT, T HH, HHT } P (A = 1/2. The basic operations of set theory have a direct probabilistic interpretation: 1

2 The event A B is the set of outcomes wich belong either to A or to B. We say that P (A B is the probability that either A or B occurs. The event A B is the set of outcomes which belong to A and to B. We say that P (A B is the probability that A and B occurs. The event A \ B is the set of outcomes which belong to A but not to B. We say that P (A \ B is the probability that A occurs but B does not occurs. The event A = S \ A is the set of outcomes which do not belong to A. We say that P (A is the probability that A does not occur. We have the following simple rules to compute probability of events. Check them! Theorem 1. Suppose A and B are events. Then we have 1. 0 P (A 1 for any event A S. 2. P (A P (B if A B. 3. P (A = 1 P (A 4. P (A B = P (A + P (B if A and B are disjoint.. P (A B = P (A + P (B P (A B for general A and B. Proof. We let the reader check 1. to 4. For. we can reduce ourselves to 4. by writing A B as the union of two disjoint event, for example We do have, by 4. A B = A (B \ A. P (A B = P (A + P (B \ A (1 On the other hand we can write B has the union of two disjoint sets (the outcomes in B which are also in A or not. B = (B A (B \ A and so by 4. So by combining (1 and (2 we find. P (B = P (B A + P (B \ A (2 Example: Tossing three coins again let A be the event that the first toss is head while B is the event that the second toss is tail. Then A = {HHH, HHT, HT H, HT T }, 2

3 and B = {HT H, HT T, T T H, T T T }, A B = {HT H, HT T }. We have P (A B = P (A + P (B P (A B = 1/2 + 1/2 1/4 = 3/4. Odds vs probabilities. Often, especially in gambling situations the randomness of the experiment is expressed in terms of odds rather than probabilities. For we make a bet at to 1 odds that U of X will beat U of Z in next week basketball game. What is meant is that the probability that X wins is thought to be times greater than the probability that Y wins. That is we have p = P (X wins = P (Y wins = P (X looses = (1 p and thus we have p = (p 1 of p = /6. More generally we have The odds of an event A are r to s P (A 1 P (A = r s P (A = r/s r/s + 1 Uniform distribution ( Naive probabilities. In many examples it is natural to assign the same probability to each event in the sample space. If the sample space is S we denote by the cardinality of S by Then for every event i S we set #S = number of elements in S p(i = 1 #S, and for any event A we have p(a = #A #S 3

4 Example. Throw two fair dice. The sample space is the set of pairs (i, j with i and j an integer between 1 and 6 and has cardinality 36. We then obtain for example P (Sum of the dice is 2 = 1 36, P (Sum of the dice is 9 = 4 36, The birthday problem. A classical problem in probability is the following. What is the probability that among N people at least 2 have the same birthday. As it turns out, and at first sight maybe surprisingly, one needs few people to have a high probability of matching birthdays. For example for N = 23 there is a probability greater than 1/2 than at least two people have the same birthday. To compute this we will make the simplifying but reasonable assumptions that there is no leap year and that every birthday is equally likely. If there are N people present, the sample space S is the set of all birthdays of the everyone. Since there is 36 choice for everyone we have We consider the event #S = 36 N A = at least two people have the same birthday It is easier to consider instead the complementary event B = A = no pair have the same birthday To compute the cardinality of B we make a list of the N people (the order does not matter. There is 36 choice of birthday for the first one on the list, for the second one on the list, there is only 364 choice of birthday if they do not have the same birthday. Continuing in the same way we find and so #B = (36 N (36 (N 1 P (B = ( 36 N = ( 1 2 ( 1 N To compute this efficiently we recall from calculus (use L Hospital rule to prove this that for any number x we have e x = lim (1 + x n n n 4

5 We take n = 36 which is a reasonably large number and take x = 1, 2,. We have the approximation ( 1 j e j/ We find then P (B 1 e 1/36 e 2/36 e (N 1/36 = e (1+2+ N 1/36 = e N(N 1/730 N(N + 1 by using the well-known identity N =. 2 How many people are needed to have a probability of 1/2 of having 2 same birthday. We have P (B = 1 2 e N(N 1/730 N(N 1 = 730 ln(2 Even for moderately small N, N(N 1 N 2 and so we find the approximate answer N 730 ln(2 = That is if there are 23 people in a room, the probability that two have the same birthday is greater than 1/2. Similarly we find that if there are N 730 ln(10 = people in the room there is a probability greater than.9 than two people have the same birthday.

6 Part 2: Combinatorics In many problems in probability where one uses uniform distribution (many of them related to gambling s one needs to count the number of outcomes compatible with a certain event. In order to do this we shall need a few basic facts of combinatorics Permutations: Suppose you have n objects and you make a list of these objects. There are n! = n(n 1(n 2 1 different way to write down this list, since there are n choices for the first on the list, n 1 choice for the second, and so on. The number n! grows very fast with n. Often it is useful to have a good estimate of n! for large n and such an estimate is given by Stirling s formula Stirling s formula n! n n e n 2πn a n where the symbols a n b n means here that lim = 1 n b n Combinations: Suppose you have a group of n objects and you wish to select a j of the n objects. The number of ways you can do this defines the binomial coefficients ( n = # of ways to pick j objects out of n objects j and this pronounced n choose j. Example: The set U = {a, b, c} has 3 elements. The subsets of U are, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} and there are ( ( 3 0 = 1 subset with 0 elements, 3 ( 1 = 3 subset with 1 elements, 3 2 = 3 subset with 2 elements, and ( 3 3 = 1 subset with 3 elements. Formulas involving binomial coefficients and Storyproofs : There are many relations between binomial coefficients. One can prove these relation using the formula for the coefficients derived a bit later. A very elegant alternative is often to use the meaning of the coefficients and to make up a story. We give a few example 6

7 1. We have the equality ( ( n n = k n k For example we have ( ( 10 6 = To see why this is true think of forming a group of k people out of n people. You can do by selecting k people with ( n k choices. Alternatively you can for the group by selecting all the people who are not among the group, that is you select n k people not in the group and there ( n n k ways of doing this. 2. Recursion relation for the binomial coefficients:and Pascal triangle: There is a simple recursion relation for the binomial coefficients ( n j in terms of the binomial coefficients ( n 1 k : ( ( ( n n 1 n 1 = + (4 j j j 1 for 0 < j < n. To use this recursion one needs to know that ( ( n 0 = n n = 1. To see why the formula (4 holds think of a group of n people. The left hand side of (4 is the number of ways to form groups of j people out of those n people. Now let ( us pick one distinguished individual among the n, let us say we pick Bob. Then n 1 j is the number of way to choose a group of j people which do not include Bob (pick j out of the remaining n 1 while ( n 1 j 1 is the number of ways to pick a group of j people which does include Bob (pick Bob and then pick j 1 out of the remaining n 1. Adding these two we obtain the right hand side of (4. (3 3. We have the relation ( ( n n 1 k = n k k 1 ( To with this holds imagine selecting a team of k (out of n and selecting also a captain for the team. Then you can either pick first the team ( nchoosek ways and then selecting the captain (k choices. This gives the left hand side of (. Alternatively you can pick first the captain (n choices and then select the rest of the team ( n 1choosek 1 ways.this gives the right hand side of (. Formula for the binomial coefficients To find an explicit formula for ( n j we note first that n(n 1 n (j 1 is the number of ways to write an ordered list of j objects out of n objects since there are n choices for the first one on the list, n 1 choices for the second one and so on. Many 7

8 of these lists contain the same objects but arranged in a different order and there are j! ways to write a list of the same j objects in different orders. So we have ( n n(n 1 n (j 1 = j j! which we can rewrite as ( n = j n! j!(n j! Poker hands. We will compute the probability of certain poker hands. A poker hands consists of a randomly chosen cards out of a deck of 2. So we have ( 2 Total number of poker hands = = Four of a kind: This hands consists of 4 cards of the same values (say 4 seven. To compute the probability of a four of a kinf note that there are 13 choices for the choice of values of the four of a kind. Then there are 48 cards left and so 48 choice for the remaining cards. So Probability of a four of a kind = ( 2 = = Full house: This hands consists three cards of the same value and two cards of an another value (e.g. 3 kings and 2 eights. There are 13 ways to choose the value of three of a kind and once this value is chose there is ( 4 3 to select the three cards out of the four of same value. There are then 12 values left to choose from for the pair and there ( 4 2 to select the the pair. So we have Probability of a four of a full house = 13 ( ( = = So the full house is 6 times as likely as the four of a kind. Three of a kind: There are 13 ( 4 3 ways to pick a three of kind. There are then 48 cards left from which to choose the remaining last 2 cards and there are ( 48 2 ways to do this. But we are then also allowing to pick a pair for the remaining two cards which would give a full house. Therefore we have Probability of a three of a kind = 13 ( 3 ( ( 3 ( ( 2 2 ( 2 8

9 Another way to compute this probability is to note that among the 48 remaining cards we should choose two different values (so as not to have a pair and then pick a card of that value. This gives Probability of a three of a kind = 13 ( 3 ( ( 1 ( ( Either way this gives a probability = Keno. This is a popular form of lottery played in casinos as well as often in bars and restaurants. For example in Massachusetts, the numbers are drawn every four minutes and appear on screens. The game is played with the numbers {1, 2, 3,, 80} and the casino randomly selects 20 numbers out of those. Clearly there are ( choices. The player plays by selecting m numbers out of 80. The number m varies and typically the game allows for the player to choose m (for example in Massachusetts any m with 1 m 12 is allowed. One say that a player gets a catch of k if k of his m numbers matches some of the 20 numbers selected by the casino. Let us take m = 8 and compute the probability of a catch of k in this case. Think now of the 80 numbers divided into two groups the 8 good numbers selected by the player and the 72 bad numbers which are not. For the player to get a catch of k, k of his numbers must be selected by the casino from his 8 good numbers and the remaining 20 k numbers are selected from the bad numbers. So we find ( 8 72 k( P ( catch of k if playing 8 numbers = k ( In general we find P ( catch of k if playing m numbers = ( m ( 80 m k k ( We will revisit the game of Keno later: the list of all odds and payouts as well as the the detailed rules for Keno as played Massachusetts can be found at com/games/keno.html. 9

10 Exercises Exercise 1: As seen in class if there are 23 people in a room the probability of having two people with the same birthday is more than 1/2. In our class of 40 people what is this probability? For comparison compute the probability that somebody has the same birthday than you in our class? Are you surprised by the result? Exercise 2: Bob and Maria are taking a math class with final grades A, B or C. The probability that Bob gets a B is.3 and the probability that Maria gets a B is.4. The probability that neither gets an A but at least one gets a B is.1. What is the probability that at least one gets a B but neither gets a C? Exercise 3: 1. What odds should you give in favor of the following event? (a A card chosen at random from a 2-card deck is an ace? (b Exactly two heads will turn up when three coins are tossed? 2. In a horse race the odds that Romance wins are given 2 to 3 while the odds that Downhill wins are 1 to 2. Give the odds that either Romance or Downhill wins? Exercise 4: A six card hand is dealt from an ordinary deck of 2 cards. Find the probability that 1. All six cards are hearts 2. There are three aces, two kings and one queen. 3. There three cards of one suit and three of another suit. Exercise : Compute the probabilities to obtain the following poker hands 1. Two pairs 2. A straight flush: fives cards of the same suit in order (e.g. 6, 7, 8, 9, 10 of hearts. 3. A flush : five cards of the same suit but not in order (e.g. 3,,6, queen,and king of spades. 10

11 Exercise 6: The powerball is a popular lottery organized by the multistage lottery association where white balls are drawn out of a drum which 9 balls and one red ball is drawn out of another drum with 3 balls. The balls are drawn without replacement and the order in which balls are drawn does not matter. The prize of the ticket is \$2 and there are 9 ways to win given in the table below. Compute the corresponding probabilities. Balls Prize Probabilities white & 1 red Jackpot white \$ 1,000,000 4 white & 1 red \$10,000 4 white \$100 3 white \$100 3 white \$7 2 white & 1red \$7 1 white & 1red \$4 1 red \$4 See for more details. Exercise 7: Explain with a story proof why the identity ( 2n = n n j=0 ( 2 n j holds. Hint: Think of a group of consisting of n boys and n girls. 11

### 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

### 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.

### 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

### 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

### 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

### 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.

### 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

### 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

### 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,

### 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

### 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 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 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

### Unit 9: Probability Assignments

Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose

### 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

### 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

### 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

### 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

### 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!

### 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

### 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

### 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.

### LAMC Junior Circle February 3, Oleg Gleizer. Warm-up

LAMC Junior Circle February 3, 2013 Oleg Gleizer oleg1140@gmail.com Warm-up Problem 1 Compute the following. 2 3 ( 4) + 6 2 Problem 2 Can the value of a fraction increase, if we add one to the numerator

### 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

### 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

### 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

### 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,

### Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Diana Pell Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 1. An experiment

### 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

### 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

### 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

### 10-1. Combinations. Vocabulary. Lesson. Mental Math. able to compute the number of subsets of size r.

Chapter 10 Lesson 10-1 Combinations BIG IDEA With a set of n elements, it is often useful to be able to compute the number of subsets of size r Vocabulary combination number of combinations of n things

### Here are two situations involving chance:

Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)

### 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:

### Class XII Chapter 13 Probability Maths. Exercise 13.1

Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:

### Diamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES

CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times

### 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,

### Multiple Choice Questions for Review

Review Questions Multiple Choice Questions for Review 1. Suppose there are 12 students, among whom are three students, M, B, C (a Math Major, a Biology Major, a Computer Science Major. We want to send

### Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Diana Pell Section 11.1: The Fundamental Counting Principle Exercise 1. How many different two-letter words (including nonsense words) can be formed when

### Grade 6 Math Circles Fall Oct 14/15 Probability

1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.

### 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

### 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

### 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

### 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

### 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

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

### TEST A CHAPTER 11, PROBABILITY

TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability

### Elementary Combinatorics

184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are

### 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

### 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 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

### 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:

### 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

### 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

### INDIAN STATISTICAL INSTITUTE

INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 20-07-07 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it

### 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.

### 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

### Probability with Set Operations. MATH 107: Finite Mathematics University of Louisville. March 17, Complicated Probability, 17th century style

Probability with Set Operations MATH 107: Finite Mathematics University of Louisville March 17, 2014 Complicated Probability, 17th century style 2 / 14 Antoine Gombaud, Chevalier de Méré, was fond of gambling

### 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.

### 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

### STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.

Worksheet 4 th Topic : PROBABILITY TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. BASIC COMPETENCY:

### Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:

Math 22 Fall 2017 Homework 2 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 1.2, Exercises 5, 7, 13, 16. Section 1.3, Exercises,

### 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

### 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

### 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

### Topics to be covered

Basic Counting 1 Topics to be covered Sum rule, product rule, generalized product rule Permutations, combinations Binomial coefficients, combinatorial proof Inclusion-exclusion principle Pigeon Hole Principle

1 of 5 7/16/2009 6:57 AM Virtual Laboratories > 13. Games of Chance > 1 2 3 4 5 6 7 8 9 10 11 3. Simple Dice Games In this section, we will analyze several simple games played with dice--poker dice, chuck-a-luck,

### 4.1 What is Probability?

4.1 What is Probability? between 0 and 1 to indicate the likelihood of an event. We use event is to occur. 1 use three major methods: 1) Intuition 3) Equally Likely Outcomes Intuition - prediction based

### 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

### 9.5 Counting Subsets of a Set: Combinations. Answers for Test Yourself

9.5 Counting Subsets of a Set: Combinations 565 H 35. H 36. whose elements when added up give the same sum. (Thanks to Jonathan Goldstine for this problem. 34. Let S be a set of ten integers chosen from

### 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

### 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

### 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

### 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

### Contents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting - Permutation and Combination 39

CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting - Permutation and Combination 39 2.5

### 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 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

### 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

### Exercise Class XI Chapter 16 Probability Maths

Exercise 16.1 Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total

### Combinatorics. Chapter Permutations. Counting Problems

Chapter 3 Combinatorics 3.1 Permutations Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and

### 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

### 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

### 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

### 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

### CISC 1400 Discrete Structures

CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest

### 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

### Well, there are 6 possible pairs: AB, AC, AD, BC, BD, and CD. This is the binomial coefficient s job. The answer we want is abbreviated ( 4

2 More Counting 21 Unordered Sets In counting sequences, the ordering of the digits or letters mattered Another common situation is where the order does not matter, for example, if we want to choose 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

### Simulations. 1 The Concept

Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that can be

### Simple Probability. Arthur White. 28th September 2016

Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and

### Poker: Probabilities of the Various Hands

Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13

### The game of poker. Gambling and probability. Poker probability: royal flush. Poker probability: four of a kind

The game of poker Gambling and probability CS231 Dianna Xu 1 You are given 5 cards (this is 5-card stud poker) The goal is to obtain the best hand you can The possible poker hands are (in increasing order):

### 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

### The student will explain and evaluate the financial impact and consequences of gambling.

What Are the Odds? Standard 12 The student will explain and evaluate the financial impact and consequences of gambling. Lesson Objectives Recognize gambling as a form of risk. Calculate the probabilities

### 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

### 1. How many subsets are there for the set of cards in a standard playing card deck? How many subsets are there of size 8?

Math 1711-A Summer 2016 Final Review 1 August 2016 Time Limit: 170 Minutes Name: 1. How many subsets are there for the set of cards in a standard playing card deck? How many subsets are there of size 8?

### 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