# MATH 105: Finite Mathematics 7-1: Sample Spaces and Assignment of Probability

Save this PDF as:

Size: px
Start display at page:

Download "MATH 105: Finite Mathematics 7-1: Sample Spaces and Assignment of Probability"

## Transcription

1 MATH 105: Finite Mathematics 7-1: Sample Spaces and Assignment of Probability Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

2 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

3 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

4 Introduction to Probability Many real world events can be considered chance or random. They may be deterministic, but we can not know or comprehend all the factors which determine the outcome. You flip a coin. Air current, the arrangement of the coin on your finger, the force of your flip, and other factors all go together to determine the outcome of Heads or Tails. For any one toss, these factors are too complicated to take into account, and the outcome appears random. Since the outcome is heads roughly half the time, we assign the following probabilities: Pr[H] = 1 2 Pr[T ] = 1 2

5 Introduction to Probability Many real world events can be considered chance or random. They may be deterministic, but we can not know or comprehend all the factors which determine the outcome. You flip a coin. Air current, the arrangement of the coin on your finger, the force of your flip, and other factors all go together to determine the outcome of Heads or Tails. For any one toss, these factors are too complicated to take into account, and the outcome appears random. Since the outcome is heads roughly half the time, we assign the following probabilities: Pr[H] = 1 2 Pr[T ] = 1 2

6 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

7 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

8 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

9 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

10 Probability Vocabulary Probability Terms Outcome A particular result of an activity or event. Event A set of outcomes which share a common characteristic. Sample Space The set of all possible outcomes for an experiment. This is the universal set for the experiment. Equally Likely Events All events in the sample space have the same probability.

11 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

12 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

13 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

14 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? S = {H, T } You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

15 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? S = {H, T } You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment?

16 Finding Sample Spaces One of the first tasks in finding probability is to determine the sample space for the experiment. You flip a fair coin. What is the sample space for this experiment? S = {H, T } You roll a six-sided die and note the number which appears on top. What is the sample space for this experiment? S = {1, 2, 3, 4, 5, 6}

17 Finding More Sample Spaces You flip a coin and roll a die, and note the result of each. what is the sample space for this experiment? S = {H1, H2,..., H6, T 1, T 2,..., T 6} c(s) = 2 6 = 12

18 Finding More Sample Spaces You flip a coin and roll a die, and note the result of each. what is the sample space for this experiment? S = {H1, H2,..., H6, T 1, T 2,..., T 6} c(s) = 2 6 = 12

19 Finding More Sample Spaces You flip a coin and roll a die, and note the result of each. what is the sample space for this experiment? S = {H1, H2,..., H6, T 1, T 2,..., T 6} c(s) = 2 6 = 12

20 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? You roll two dice and note the sum of the two numbers. What is the sample space for this experiment?

21 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? S = {(1, 1), (1, 2),..., (2, 1), (2, 2),...} c(s) = 6 6 = 36 You roll two dice and note the sum of the two numbers. What is the sample space for this experiment?

22 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? S = {(1, 1), (1, 2),..., (2, 1), (2, 2),...} c(s) = 6 6 = 36 You roll two dice and note the sum of the two numbers. What is the sample space for this experiment?

23 Different Sample Spaces for the Same Experiment You roll two dice and note both numbers. What is the sample space for this experiment? S = {(1, 1), (1, 2),..., (2, 1), (2, 2),...} c(s) = 6 6 = 36 You roll two dice and note the sum of the two numbers. What is the sample space for this experiment? S = {2, 3,..., 12} c(s) = 11

24 Taking a Quiz You take a True/False quiz with three questions. If you treat this quiz as an experiment, what is the sample space?

25 Taking a Quiz You take a True/False quiz with three questions. If you treat this quiz as an experiment, what is the sample space? S = {TTT, TTF,..., FFF } c(s) = 8

26 Taking a Quiz You take a True/False quiz with three questions. If you treat this quiz as an experiment, what is the sample space? S = {TTT, TTF,..., FFF } c(s) = 8 Now that we have some practice identifying sample spaces, it is time to start assigning probabilities.

27 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

28 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

29 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? S = {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF } Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

30 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? S = {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF } A few rules before we actually assign probabilities. Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

31 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? S = {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF } A few rules before we actually assign probabilities. Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

32 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? S = {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF } A few rules before we actually assign probabilities. Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

33 Taking a Quiz How likely are you to get all three answers in the True/False quiz correct if you guess on each question? S = {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF } A few rules before we actually assign probabilities. Rules for Assigning Probability For each outcome W, 0 Pr[W ] 1 The sum of the probabilities of all outcomes is one. Equally Likely Outcomes Pr[TTT ] = Pr[TTF ] =... = Pr[FFF ] = 1 c(s) = 1 8

34 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

35 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

36 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

37 Probability Model When you find the sample space for an experiment and assign probabilities to each element of the sample space, you are constructing a probability model. A six sided die is weighted so that the 1 is twice as likely as any other number and all other numbers are equally likely. Find the probability model. S = { 1, 2, 3, 4, 5, 6, } 2x x x x x x Pr[1] = 2 7 Pr[2] = Pr[3] = Pr[4] = Pr[5] = Pr[6] = 1 7

38 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You guess on all 3 questions in the True/False quiz seen earlier. What is the probability that you miss one? E = {TTF, TFT, FTT } Pr[E] = c(e) c(s) = 3 8

39 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You guess on all 3 questions in the True/False quiz seen earlier. What is the probability that you miss one? E = {TTF, TFT, FTT } Pr[E] = c(e) c(s) = 3 8

40 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You guess on all 3 questions in the True/False quiz seen earlier. What is the probability that you miss one? E = {TTF, TFT, FTT } Pr[E] = c(e) c(s) = 3 8

41 Probabilities of Events To find the probability of an event in sample spaces with equally likely outcomes, we use the following probability formula. Probability of an Event If E is a subset of a sample space S in which all outcomes are equally likely, then Pr[E] = c(e) c(s) You guess on all 3 questions in the True/False quiz seen earlier. What is the probability that you miss one? E = {TTF, TFT, FTT } Pr[E] = c(e) c(s) = 3 8

42 Drawing Balls from an Urn A jar contains 8 balls: 4 green, 3 blue, and 1 red. You pick one ball at random. Find: 1 The probability the ball you draw is green. 2 The probability the ball you draw is not red. An urn contains 3 balls: one red, one green, and one yellow. You draw the balls out one-by-one at random. What is the probability that the yellow ball is not drawn drawn last?

43 Drawing Balls from an Urn A jar contains 8 balls: 4 green, 3 blue, and 1 red. You pick one ball at random. Find: 1 The probability the ball you draw is green. 2 The probability the ball you draw is not red. An urn contains 3 balls: one red, one green, and one yellow. You draw the balls out one-by-one at random. What is the probability that the yellow ball is not drawn drawn last?

44 Rolling Two Dice You roll two fair six-sided dice and note the sum of the rolls. Find each probability. 1 Pr[ sum is 7 ] 2 Pr[ sum is 4 ] 3 Pr[ sum is 4 or 7 ] 4 Pr[ sum is 4 and 7 ]

45 Outline 1 Probability 2 Sample Spaces 3 Assigning Probability 4 Conclusion

46 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

47 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

48 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

49 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

50 Important Concepts Things to Remember from Section Probability Vocabulary: Outcomes, Events, Sample Spaces 2 Finding Sample Spaces 3 Building Probability Models 4 Assigning Probabilities to Events

51 Next Time... Since probabilities are based on sets: the sample space and events, it is conceivable that tools used to work with sets would also be important in working with probabilities. Indeed, next time we will use rules for combining sets and Venn Diagrams to help solve probability problems. For next time Read Section 7-2 (pp ) Do Problem Sets 7-1 A,B

52 Next Time... Since probabilities are based on sets: the sample space and events, it is conceivable that tools used to work with sets would also be important in working with probabilities. Indeed, next time we will use rules for combining sets and Venn Diagrams to help solve probability problems. For next time Read Section 7-2 (pp ) Do Problem Sets 7-1 A,B

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

### Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### https://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10

1 of 8 4/9/2013 8:17 AM PRINTABLE VERSION Quiz 10 Question 1 Let A and B be events in a sample space S such that P(A) = 0.34, P(B) = 0.39 and P(A B) = 0.19. Find P(A B). a) 0.4872 b) 0.5588 c) 0.0256 d)

### Question of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay

QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know

### Section 6.2 Definition of Probability

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 that it will

### Math 3C Homework 3 Solutions

Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard

### Basic Probability Theory I

A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population

### Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

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

Ch. - Problems to look at Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability

### Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com

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..

### Chapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.

Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive

### ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers

ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### ACMS 10140 Section 02 Elements of Statistics October 28, 2010. Midterm Examination II

ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages of the examination

### 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.

Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### Mathematical goals. Starting points. Materials required. Time needed

Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

### Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

### Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin. You

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Chapter 4 - Practice Problems 1

Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula

### STAT 35A HW2 Solutions

STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

### Statistics 100A Homework 2 Solutions

Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

### Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.

Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers

### MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

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

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)

Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities

### The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

### Grade 6 Math Circles March 2, 2011 Counting

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 6 Math Circles March 2, 2011 Counting Venn Diagrams Example 1: Ms. Daly surveyed her class of 26

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Probabilistic Strategies: Solutions

Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

### Ch. 12.1: Permutations

Ch. 12.1: Permutations The Mathematics of Counting The field of mathematics concerned with counting is properly known as combinatorics. Whenever we ask a question such as how many different ways can we

### For two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B)

Basic probability A probability space or event space is a set Ω together with a probability measure P on it. This means that to each subset A Ω we associate the probability P(A) = probability of A with

### Introduction to Probability

Introduction to Probability Math 30530, Section 01 Fall 2012 Homework 1 Solutions 1. A box contains four candy bars: two Mars bars, a Snickers and a Kit-Kat. I randomly draw a bar from the box and eat

### Worked examples Basic Concepts of Probability Theory

Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one

### Introduction to Probability

3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which

### Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

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

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

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### 36 Odds, Expected Value, and Conditional Probability

36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face

### Basic Probability Theory II

RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample

### Formula for Theoretical Probability

Notes Name: Date: Period: Probability I. Probability A. Vocabulary is the chance/ likelihood of some event occurring. Ex) The probability of rolling a for a six-faced die is 6. It is read as in 6 or out

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

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

### Probability OPRE 6301

Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

### Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments

### Math 210. 1. Compute C(1000,2) (a) 499500. (b) 1000000. (c) 2. (d) 999000. (e) None of the above.

Math 210 1. Compute C(1000,2) (a) 499500. (b) 1000000. (c) 2. (d) 999000. 2. Suppose that 80% of students taking calculus have previously had a trigonometry course. Of those that did, 75% pass their calculus

### STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia

STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that

### Number of events classifiable as A Total number of possible events

PROBABILITY EXERCISE For the following probability practice questions, use the following formulas. NOTE: the formulas are in the basic format and may require slight modification to account for subsequent

### Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR.

Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR. 1. Urn A contains 6 white marbles and 4 red marbles. Urn B contains 3 red marbles and two white

### (b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.

Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw

### Chapter 4 - Practice Problems 2

Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### MATH 105: Finite Mathematics 6-5: Combinations

MATH 105: Finite Mathematics 6-5: Combinations Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Developing Combinations 2 s of Combinations 3 Combinations vs. Permutations 4 Conclusion

### 4.4 Conditional Probability

4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Exam Name 1) Solve the system of linear equations: 2x + 2y = 1 3x - y = 6 2) Consider the following system of linear inequalities. 5x + y 0 5x + 9y 180 x + y 5 x 0, y 0 1) 2) (a) Graph the feasible set

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### Mathematical Expectation

Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the

### Chapter 13 & 14 - Probability PART

Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph

### Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2

Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability

### STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012)

STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012) TA: Zhen (Alan) Zhang zhangz19@stt.msu.edu Office hour: (C500 WH) 1:45 2:45PM Tuesday (office tel.: 432-3342) Help-room: (A102 WH) 11:20AM-12:30PM,

### PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

### Homework 3 Solution, due July 16

Homework 3 Solution, due July 16 Problems from old actuarial exams are marked by a star. Problem 1*. Upon arrival at a hospital emergency room, patients are categorized according to their condition as

### 6th Grade Lesson Plan: Probably Probability

6th Grade Lesson Plan: Probably Probability Overview This series of lessons was designed to meet the needs of gifted children for extension beyond the standard curriculum with the greatest ease of use

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant

### Chapter 3 Random Variables and Probability Distributions

Math 322 Probabilit and Statistical Methods Chapter 3 Random Variables and Probabilit Distributions In statistics we deal with random variables- variables whose observed value is determined b chance. Random

### ECE302 Spring 2006 HW1 Solutions January 16, 2006 1

ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### CLAT Junction Maths Exercise

CLAT Junction Maths Exercise 16 www.clatjunction.com clatjunction@gmail.com 9472474746 Maths Exercise 16 1. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

### Combinations and Permutations

Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination

### Notes on Probability. Peter J. Cameron

Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some

### Statistics 100A Homework 3 Solutions

Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win \$ for each black ball selected and we

### Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical

### Probabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I

Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with

### AP Statistics 7!3! 6!

Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### 1 Combinations, Permutations, and Elementary Probability

1 Combinations, Permutations, and Elementary Probability Roughly speaking, Permutations are ways of grouping things where the order is important. Combinations are ways of grouping things where the order

### PROBABILITY 14.3. section. The Probability of an Event

4.3 Probability (4-3) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques

### CHAPTER 3: PROBABILITY TOPICS

CHAPTER 3: PROBABILITY TOPICS Exercise 1. In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities

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

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

### Hoover High School Math League. Counting and Probability

Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches

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

Great Theoretical Ideas In Computer Science Victor Adamchi CS 5-5 Carnegie Mellon University Probability - I The Story The theory of probability was originally developed by a French mathematician Blaise

### The Casino Lab STATION 1: CRAPS

The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

### Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

### Probability Using Dice

Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you