# MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG

Size: px
Start display at page:

Transcription

1 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 on each floor. How many offices are in the building? (b A particular brand of shirt comes in colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?. There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?. (a How many different three-letter initials can people have? (b How many different three-letter initials with none of the letters repeated can people have?. (a How many bit strings of length ten both begin and end with a? (b How many bit strings are there of length six or less, not counting the empty string? (c How many bit strings of length n, where n is a positive integer, start and end with s?. How many strings are there of four lowercase letters that have the letter x in them?. (a How many license plates can be made using either three digits followed by three uppercase English letters or three uppercase English letters followed by three digits? (b How many license plates can be made using either three uppercase English letters followed by three digits or four uppercase English letters followed by two digits? 7. (a How many subsets of a set with elements have more than one element? (b A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes? 8. (a Use the principle of inclusion-exclusion to find the number of positive integers less than,, that are not divisible by either or by. (b A wired equivalent privacy (WEP key for a wireless fidelity (WiFi network is a string of either,, or 8 hexadecimal digits. How many different WEP keys are there? (There are place values for hexadecimal numbers: to 9, A, B, C, D, E, and F. Permutations and Subsets 9. Let S = {,,,, }. (a List all the -permutations of S. (b List all the -subsets of S.. Find the value of each of these quantities. (a P (, (b P (, (c P (8,

2 (d P (,. Find ( the value of each of these ( quantities. (a (b (c (d (. In how many different orders can five runners finish a race if no ties are allowed?. There are six different candidates for governor of a state. In how many different orders can the names of the candidates be printed on a ballot?. How many bit strings of length contain (a exactly three s? (b at most three s? (c at least three s? (d an equal number of s and s?. In how many ways can a set of two positive integers less than be chosen?. Suppose that there are 9 faculty members in the mathematics department and in the computer science department. How many ways are there to form a basketball team that has math professors and computer science professors? 7. A coin is flipped eight times where each flip comes up either heads or tails. How many possible outcomes (a are there in total? (b contain exactly three heads? (c contain at least three heads? (d contain the same number of heads and tails? 8. Thirteen people on a softball team show up for a game. (a How many ways are there to choose players to take the field? (b How many ways are there to assign the positions by selecting players from the people who show up? (c Of the people who show up, three are women. How many ways are there to choose players to take the field if at least one of these players must be a woman? 9. A club has members. (a How many ways are there to choose four members of the club to serve on an executive committee? (b How many ways are there to choose a president, vice president, secretary, and treasurer of the club, where no person can hold more than one office?. Suppose that a department contains men and women. How many ways are there to form a committee with six members if it must have the same number of men and women?

3 Pigeonhole Principle. (a Show that if there are students in a class, then at least two have last names that begin with the same letter. (b What is the minimum number of students, each of whom comes from one of the states, who must be enrolled in a university to guarantee that there are at least who come from the same state?. A bowl contains red balls and blue balls. A woman selects balls at random without looking at them. (a How many balls must she select to be sure of having at least three balls of the same color? (b How many balls must she select to be sure of having at least three blue balls?. Assuming that no one has more than,, hairs on the head of any person and that the population of New York City was 8, 8, 78 in, show there had to be at least nine people in New York City in with the same number of hairs on their heads. Uniform Discrete Distributions. (a What is the probability that a card selected at random from a standard deck of cards is an ace? (b What is the probability that a fair die comes up six when it is rolled? (c What is the probability that a randomly selected integer chosen from the first positive integers is odd? (d What is the probability that a randomly selected day of a leap year (with possible days is in April?. (a What is the probability that a five-card poker hand contains the ace of hearts? (b What is the probability that a five-card poker hand does not contain the queen of hearts? (c What is the probability that a five-card poker hand contains the two of diamonds and the three of spades? (d What is the probability that a five-card poker hand contains the two of diamonds, the three of spades, the six of hearts, the ten of clubs, and the king of hearts? (e What is the probability that a five-card poker hand contains exactly one ace?. (a What is the probability that a fair die never comes up an even number when it is rolled six times? (b What is the probability that a positive integer not exceeding selected at random is divisible by? 7. In a super lottery, players win a fortune if they choose the eight numbers selected by a computer from the positive integers not exceeding. What is the probability that a player wins this super lottery? Distributions and Probability of Unions and Complements 8. Find the probability of each outcome when a biased die is rolled, if rolling a or rolling a is three times as likely as rolling each of the other four numbers on the die and it is equally likely to roll a or a.

4 9. What probability should be assigned to the outcome of heads when a biased coin is tossed, if heads is three times as likely to come up as tails? What probability should be assigned to the outcome of tails?. A pair of dice is loaded. The probability that a appears on the first die is /7, and the probability that a appears on the second die is /7. Other outcomes for each die appear with probability /7. What is the probability of 7 appearing as the sum of the numbers when the two dice are rolled?. Assume that the year has days and all birthdays are equally likely. (a Find the smallest number of people you need to choose at random so that the probability that at least one of them has a birthday today exceeds /. (b Find the smallest number of people you need to choose at random so that the probability that everyone has a distinct birthday is below /.. Assume that the probability a child is a boy is. and that the sexes of children born into a family are independent. What is the probability that a family of five children has... (a exactly three boys? (c at least one girl? (b at least one boy? (d all children of the same sex? Conditional Probability and Bayes Theorem. A wedding party of eight people is lined up in a random order. (a What is the probability that the bride is next to the groom? (b What is the probability that maid of honor is in the leftmost position? (c Determine whether the two events are independent. Give your reasoning.. A red die and a blue die are thrown. Define the following events: A: The sum is even. B: The sum is at least. C: The red die comes up. Find the following probabilities. (a P r(a, (b P r(b, (c P r(c (d P r(a C (e P r(b C, (f P r(a B.. Sally has two coins. The first coin is a fair coin and the second coin is biased. The biased coin comes up heads with probability.7 and tails with probability.. She selects a coin at random and flips the coin ten times. Out of the ten coin flips, 7 flips come up heads and come up tails. What is the probability that she selected the biased coin?. Assume that you have two dice, one of which is fair, and the other is biased toward landing on six, so that of the time it lands on six, and of the time it lands on each of,, and, and of the time on. You choose a die at random, and spin it six times, getting the values,,,,,. What is the probability that the die you chose is the fair die? 7. Assume one person out of, is infected with HIV, and there is a test in which.% of all people test positive for the virus although they do not really have it. If you test negative on this test, then you definitely do not have HIV. Let H be the event of having HIV and T be the event of testing positive. Find the following. (a Pr(T H, the probability of testing positive for someone with HIV.

5 (b Pr(H T, the probability of having HIV and testing positive. (c Pr(T H, the probability of testing positive for someone without HIV. (d Pr(H T, the probability of not having HIV and testing positive. (e Pr(T, the probability of testing positive. (f Pr(H T, the probability of having HIV for someone who tests positive. 8. In Small Town MN, % of residents are teenagers. 9% of teenagers use Facebook, while only % of the rest of the town use Facebook. Let T be the set of teenagers and F be the set of Facebook users. (a Find Pr(F T, the probability that a resident is a teenager and a Facebook user. (b Find Pr(F T, the probability that a resident is a non-teenager and a Facebook user. (c Find Pr(F, the probability that a resident is a Facebook user. (d Find Pr(T F, the probability for a Facebook user to be a teenager.

6 Solutions. (a 7 7 (b.. (a (b. (a 8 (b (c string when n = ; n strings when n... (a (b + 7. (a (b n/ when n is even; (n+/ when n is odd. 8. (a ( (b Skip. (a (b (c (d 9 =!. (a (b (c (d. P (, =!. P (, =! (. (a ( (c ( 99.. ( 9 ( 7. (a 8 (b ( 8. (a ( 9. (a (. ( ( (b ( ( + (b P (, (b P (, ( (c 8 (c ( + (d ( ( ( + ( (d. (a = (b (a + = (b + = = 9.. (a (b (c (d

7 (. (a ( (b ( ( (c ( ( (d ( (e ( (. (a ( (b 7. ( 8 8. P ( = P ( =., P ( = P ( = P ( = P ( =. 9. P (Head = /, P (T ail = / (a Solve n for ( n > (b Solve n for ( ( ( (n <.. (a. (b.9 (c. (d. +.9 P (, P (7,7. (a P (8,8 = (b 8 P (, P (, (c No. P (bride-next-to-groom maid-of-honor-in-leftmost = P (8,8 = (a (b (c (d (e (f.,, + 9,999,.% 7

### 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,

### 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

### 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

### 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

### Exercises Exercises. 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}?

Exercises Exercises 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}? 3. How many permutations of {a, b, c, d, e, f, g} end with

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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!

### 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

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

Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is

### Fundamentals of Probability

Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

### 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

### 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. 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

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

More 9.-9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on

### 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:

### Conditional Probability Worksheet

Conditional Probability Worksheet P( A and B) P(A B) = P( B) Exercises 3-6, compute the conditional probabilities P( AB) and P( B A ) 3. P A = 0.7, P B = 0.4, P A B = 0.25 4. P A = 0.45, P B = 0.8, P A

### Conditional Probability Worksheet

Conditional Probability Worksheet EXAMPLE 4. Drug Testing and Conditional Probability Suppose that a company claims it has a test that is 95% effective in determining whether an athlete is using a steroid.

### 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

### 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:

### 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

### 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.

### 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

### 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.

### 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

### Unit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)

Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,

### 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

### 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

### 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

### 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

### Fundamental Counting Principle

Lesson 88 Probability with Combinatorics HL2 Math - Santowski Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more

### 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

### 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

### 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 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

### 6) A) both; happy B) neither; not happy C) one; happy D) one; not happy

MATH 00 -- PRACTICE TEST 2 Millersville University, Spring 202 Ron Umble, Instr. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all natural

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

6. Practice Problems Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability. ) A bag contains red marbles, blue marbles, and 8

### The point value of each problem is in the left-hand margin. You must show your work to receive any credit, except on problems 1 & 2. Work neatly.

Introduction to Statistics Math 1040 Sample Exam II Chapters 5-7 4 Problem Pages 4 Formula/Table Pages Time Limit: 90 Minutes 1 No Scratch Paper Calculator Allowed: Scientific Name: The point value of

### 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

### 6.1 Basics of counting

6.1 Basics of counting CSE2023 Discrete Computational Structures Lecture 17 1 Combinatorics: they study of arrangements of objects Enumeration: the counting of objects with certain properties (an important

### 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

### 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

### 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

### 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:

### Finite Math B, Chapter 8 Test Review Name

Finite Math B, Chapter 8 Test Review Name Evaluate the factorial. 1) 6! A) 720 B) 120 C) 360 D) 1440 Evaluate the permutation. 2) P( 10, 5) A) 10 B) 30,240 C) 1 D) 720 3) P( 12, 8) A) 19,958,400 B) C)

### 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

### 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

### Contemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Math 1030 Sample Exam I Chapters 13-15 Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin.

### MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE

MATH 2053 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING 2009 - DR. DAVID BRIDGE MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the

### Probability. The Bag Model

Probability The Bag Model Imagine a bag (or box) containing balls of various kinds having various colors for example. Assume that a certain fraction p of these balls are of type A. This means N = total

### Name: Section: Date:

WORKSHEET 5: PROBABILITY Name: Section: Date: Answer the following problems and show computations on the blank spaces provided. 1. In a class there are 14 boys and 16 girls. What is the probability of

### 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

### 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.

### STATISTICAL COUNTING TECHNIQUES

STATISTICAL COUNTING TECHNIQUES I. Counting Principle The counting principle states that if there are n 1 ways of performing the first experiment, n 2 ways of performing the second experiment, n 3 ways

### Homework Set #1. 1. The Supreme Court (9 members) meet, and all the justices shake hands with each other. How many handshakes are there?

Homework Set # Part I: COMBINATORICS (follows Lecture ). The Supreme Court (9 members) meet, and all the justices shake hands with each other. How many handshakes are there? 2. A country has license plates

### LC OL Probability. ARNMaths.weebly.com. As part of Leaving Certificate Ordinary Level Math you should be able to complete the following.

A Ryan LC OL Probability ARNMaths.weebly.com Learning Outcomes As part of Leaving Certificate Ordinary Level Math you should be able to complete the following. Counting List outcomes of an experiment Apply

### 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

### Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11

Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value

### 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

### 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.

### MAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions

MAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions 1. Appetizers: Salads: Entrées: Desserts: 2. Letters: (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U,

### 6. In how many different ways can you answer 10 multiple-choice questions if each question has five choices?

Pre-Calculus Section 4.1 Multiplication, Addition, and Complement 1. Evaluate each of the following: a. 5! b. 6! c. 7! d. 0! 2. Evaluate each of the following: a. 10! b. 20! 9! 18! 3. In how many different

### Most of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected.

AFM Unit 7 Day 3 Notes Theoretical vs. Experimental Probability Name Date Definitions: Experiment: process that gives a definite result Outcomes: results Sample space: set of all possible outcomes Event:

### 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.

### Independent and Mutually Exclusive Events

Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A

### CPCS 222 Discrete Structures I Counting

King ABDUL AZIZ University Faculty Of Computing and Information Technology CPCS 222 Discrete Structures I Counting Dr. Eng. Farag Elnagahy farahelnagahy@hotmail.com Office Phone: 67967 The Basics of counting

### 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

### 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

### 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

### 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

### 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

### Pan (7:30am) Juan (8:30am) Juan (9:30am) Allison (10:30am) Allison (11:30am) Mike L. (12:30pm) Mike C. (1:30pm) Grant (2:30pm)

STAT 225 FALL 2012 EXAM ONE NAME Your Section (circle one): Pan (7:30am) Juan (8:30am) Juan (9:30am) Allison (10:30am) Allison (11:30am) Mike L. (12:30pm) Mike C. (1:30pm) Grant (2:30pm) Grant (3:30pm)

### Independent Events. If we were to flip a coin, each time we flip that coin the chance of it landing on heads or tails will always remain the same.

Independent Events Independent events are events that you can do repeated trials and each trial doesn t have an effect on the outcome of the next trial. If we were to flip a coin, each time we flip that

### 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

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

Mathematical Ideas Chapter 2 Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) In one town, 2% of all voters are Democrats. If two voters

### (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

### 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

### Math 1101 Combinations Handout #17

Math 1101 Combinations Handout #17 1. Compute the following: (a) C(8, 4) (b) C(17, 3) (c) C(20, 5) 2. In the lottery game Megabucks, it used to be that a person chose 6 out of 36 numbers. The order of

### COUNTING TECHNIQUES. Prepared by Engr. JP Timola Reference: Discrete Math by Kenneth H. Rosen

COUNTING TECHNIQUES Prepared by Engr. JP Timola Reference: Discrete Math by Kenneth H. Rosen COMBINATORICS the study of arrangements of objects, is an important part of discrete mathematics. Counting Introduction

### Math141_Fall_2012 ( Business Mathematics 1) Week 7. Dr. Marco A. Roque Sol Department of Mathematics Texas A&M University

( Business Mathematics 1) Week 7 Dr. Marco A. Roque Department of Mathematics Texas A&M University In this sections we will consider two types of arrangements, namely, permutations and combinations a.

### 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

### Math 146 Statistics for the Health Sciences Additional Exercises on Chapter 3

Math 46 Statistics for the Health Sciences Additional Exercises on Chapter 3 Student Name: Find the indicated probability. ) If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH

### 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,

### 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

### Chapter 3: PROBABILITY

Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of

### Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1

Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..

### MTH 103 H Final Exam. 1. I study and I pass the course is an example of a. (a) conjunction (b) disjunction. (c) conditional (d) connective

MTH 103 H Final Exam Name: 1. I study and I pass the course is an example of a (a) conjunction (b) disjunction (c) conditional (d) connective 2. Which of the following is equivalent to (p q)? (a) p q (b)

### 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

### 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)

### Section 7.1 Experiments, Sample Spaces, and Events

Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light.