# Lesson 4: Chapter 4 Sections 1-2

Size: px
Start display at page:

Download "Lesson 4: Chapter 4 Sections 1-2"

Transcription

1 Lesson 4: Chapter 4 Sections 1-2 Caleb Moxley BSC Mathematics 14 September 15

2 4.1 Randomness What s randomness?

3 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual outcomes are uncertain (in some sense) but nonetheless display a regular distribution of outcomes when trails are repeated a large number of times.

4 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual outcomes are uncertain (in some sense) but nonetheless display a regular distribution of outcomes when trails are repeated a large number of times. Example (deck of cards) Imagine drawing a card from a 52-card deck.

5 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual outcomes are uncertain (in some sense) but nonetheless display a regular distribution of outcomes when trails are repeated a large number of times. Example (deck of cards) Imagine drawing a card from a 52-card deck. What card would you draw?

6 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual outcomes are uncertain (in some sense) but nonetheless display a regular distribution of outcomes when trails are repeated a large number of times. Example (deck of cards) Imagine drawing a card from a 52-card deck. What card would you draw? If you repeated this process 52 times, how often would you expect to have drawn an ace?

7 4.1 Randomness When the same random phenomenon is repeated a large number of times, we start to see the pattern/distribution that arises among the possible outcomes.

8 4.1 Randomness When the same random phenomenon is repeated a large number of times, we start to see the pattern/distribution that arises among the possible outcomes. Definition (probability) The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

9 4.1 Randomness When the same random phenomenon is repeated a large number of times, we start to see the pattern/distribution that arises among the possible outcomes. Definition (probability) The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Example (deck of cards) What s the probability of drawing an ace from a 52-card deck?

10 4.1 Randomness When the same random phenomenon is repeated a large number of times, we start to see the pattern/distribution that arises among the possible outcomes. Definition (probability) The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. Example (deck of cards) What s the probability of drawing an ace from a 52-card deck? You often can calculate the probability of an outcome or event if you make assumptions about the phenomenon rather than having to perform a long series of repetitions.

11 4.1 Randomness In determining probabilities based on a long series of repetitions, it s important that the series be very long.

12 4.1 Randomness In determining probabilities based on a long series of repetitions, it s important that the series be very long. People often underestimate how many times you d need to repeat the phenomenon before the probabilities converge to their true probabilities.

13 4.1 Randomness In determining probabilities based on a long series of repetitions, it s important that the series be very long. People often underestimate how many times you d need to repeat the phenomenon before the probabilities converge to their true probabilities. This notion is completely analogous to the need to take large samples to get a reliable estimate of a population parameter based on a sample statistic.

14 4.1 Randomness In determining probabilities based on a long series of repetitions, it s important that the series be very long. People often underestimate how many times you d need to repeat the phenomenon before the probabilities converge to their true probabilities. This notion is completely analogous to the need to take large samples to get a reliable estimate of a population parameter based on a sample statistic. 1 Performing many trials can be difficult, so estimating probabilities or calculating them using mathematical assumptions is often necessary.

15 4.1 Randomness In determining probabilities based on a long series of repetitions, it s important that the series be very long. People often underestimate how many times you d need to repeat the phenomenon before the probabilities converge to their true probabilities. This notion is completely analogous to the need to take large samples to get a reliable estimate of a population parameter based on a sample statistic. 1 Performing many trials can be difficult, so estimating probabilities or calculating them using mathematical assumptions is often necessary. 2 Trials must always be independent, i.e the outcomes of one trail must not depend on the outcome of any others.

16 4.1 Randomness In determining probabilities based on a long series of repetitions, it s important that the series be very long. People often underestimate how many times you d need to repeat the phenomenon before the probabilities converge to their true probabilities. This notion is completely analogous to the need to take large samples to get a reliable estimate of a population parameter based on a sample statistic. 1 Performing many trials can be difficult, so estimating probabilities or calculating them using mathematical assumptions is often necessary. 2 Trials must always be independent, i.e the outcomes of one trail must not depend on the outcome of any others. 3 Often we can imitate random behavior rather than actually performing the trials. We call this simulation.

17 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries.

18 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like

19 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like gambling,

20 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like gambling, natural/environmental random phenomena,

21 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like gambling, natural/environmental random phenomena, mortality,

22 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like gambling, natural/environmental random phenomena, mortality, traffic/queuing theory,

23 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like gambling, natural/environmental random phenomena, mortality, traffic/queuing theory, ergodicity/particle movement/brownian motion,

24 4.1 Randomness A rigorous study of games of chance wasn t realized until the 16th and 17th Centuries. Since then, the study of probability has grown to a discipline that helps us understand things like gambling, natural/environmental random phenomena, mortality, traffic/queuing theory, ergodicity/particle movement/brownian motion, finance, and much more.

25 4.1 Randomness Mastery Question: Example (independence) Are the following independent trails?

26 4.1 Randomness Mastery Question: Example (independence) Are the following independent trails? The number of holiday cards you receive over a period of ten years.

27 4.1 Randomness Mastery Question: Example (independence) Are the following independent trails? The number of holiday cards you receive over a period of ten years. The temperature at Vulcan on January 1st each year for three years.

28 4.1 Randomness Mastery Question: Example (independence) Are the following independent trails? The number of holiday cards you receive over a period of ten years. The temperature at Vulcan on January 1st each year for three years. The results of five coin tosses where the coin is 40% likely to land on heads and 60% likely to land on tails.

29 4.1 Randomness Mastery Question: Example (random) Are the following random phenomena? If so, explain how a probability applies to them. The number of cars a particular family owns.

30 4.1 Randomness Mastery Question: Example (random) Are the following random phenomena? If so, explain how a probability applies to them. The number of cars a particular family owns. Whether or not a medical student passes her board exams.

31 4.1 Randomness Mastery Question: Example (random) Are the following random phenomena? If so, explain how a probability applies to them. The number of cars a particular family owns. Whether or not a medical student passes her board exams. The numbers selected in a lottery drawing.

32 A probability model consists of a list of possible outcomes and the corresponding probability of each outcome.

33 A probability model consists of a list of possible outcomes and the corresponding probability of each outcome. This two-part description fully characterizes a phenomenon s randomness.

34 A probability model consists of a list of possible outcomes and the corresponding probability of each outcome. This two-part description fully characterizes a phenomenon s randomness. Definition (outcome) An outcome of a random phenomenon is a possible result from a trail which cannot be broken down into simpler results.

35 A probability model consists of a list of possible outcomes and the corresponding probability of each outcome. This two-part description fully characterizes a phenomenon s randomness. Definition (outcome) An outcome of a random phenomenon is a possible result from a trail which cannot be broken down into simpler results. Are the following outcomes of rolling a six-sided die or not?

36 A probability model consists of a list of possible outcomes and the corresponding probability of each outcome. This two-part description fully characterizes a phenomenon s randomness. Definition (outcome) An outcome of a random phenomenon is a possible result from a trail which cannot be broken down into simpler results. Are the following outcomes of rolling a six-sided die or not? rolling a three rolling an odd number rolling an even number less than four

37 Definition (sample space) The sample space of a random phenomenon is the set of all possible outcomes.

38 Definition (sample space) The sample space of a random phenomenon is the set of all possible outcomes. Definition (event) An event is an outcome, a set of outcomes, or the entire collection of all possible outcomes of a random phenomenon. That is, it s any subset of the sample space - possibly the whole sample space.

39 Definition (sample space) The sample space of a random phenomenon is the set of all possible outcomes. Definition (event) An event is an outcome, a set of outcomes, or the entire collection of all possible outcomes of a random phenomenon. That is, it s any subset of the sample space - possibly the whole sample space. Example (number of children) A population of four families have 0, 2, 2, 1 children in each family. If we selected one family from this population at random (with equal likelihood), what is the sample space for this random trail?

40 Definition (sample space) The sample space of a random phenomenon is the set of all possible outcomes. Definition (event) An event is an outcome, a set of outcomes, or the entire collection of all possible outcomes of a random phenomenon. That is, it s any subset of the sample space - possibly the whole sample space. Example (number of children) A population of four families have 0, 2, 2, 1 children in each family. If we selected one family from this population at random (with equal likelihood), what is the sample space for this random trail? What are the outcomes?

41 Definition (sample space) The sample space of a random phenomenon is the set of all possible outcomes. Definition (event) An event is an outcome, a set of outcomes, or the entire collection of all possible outcomes of a random phenomenon. That is, it s any subset of the sample space - possibly the whole sample space. Example (number of children) A population of four families have 0, 2, 2, 1 children in each family. If we selected one family from this population at random (with equal likelihood), what is the sample space for this random trail? What are the outcomes? What is the set of all possible events?

42 The following facts are true for all probabilities and all probability models.

43 The following facts are true for all probabilities and all probability models. Any probability is a number between 0 and 1.

44 The following facts are true for all probabilities and all probability models. Any probability is a number between 0 and 1. The event containing all possible outcomes (i.e. the entire sample space) must have a probability equal to 1.

45 The following facts are true for all probabilities and all probability models. Any probability is a number between 0 and 1. The event containing all possible outcomes (i.e. the entire sample space) must have a probability equal to 1. If two events are disjoint (i.e. if they have no outcomes in common), then the probability that one or the other occurs is the sum of their individual probabilities.

46 The following facts are true for all probabilities and all probability models. Any probability is a number between 0 and 1. The event containing all possible outcomes (i.e. the entire sample space) must have a probability equal to 1. If two events are disjoint (i.e. if they have no outcomes in common), then the probability that one or the other occurs is the sum of their individual probabilities. The probability that an event does not occur is 1 minus the probability that the event does occur.

47 The following facts are true for all probabilities and all probability models. Any probability is a number between 0 and 1. The event containing all possible outcomes (i.e. the entire sample space) must have a probability equal to 1. If two events are disjoint (i.e. if they have no outcomes in common), then the probability that one or the other occurs is the sum of their individual probabilities. The probability that an event does not occur is 1 minus the probability that the event does occur. We summarize these facts in a table on the next slide.

48 Rule 1 0 P(A) 1 Rule 2 P(S) = 1 Rule 3 P(A) + P(B) = P(A or B) if A and B are disjoint Rule 4 P(A c ) = 1 P(A)

49 Rule 1 0 P(A) 1 Rule 2 P(S) = 1 Rule 3 P(A) + P(B) = P(A or B) if A and B are disjoint Rule 4 P(A c ) = 1 P(A) It s important to understand how these identities arise, and looking at Venn diagrams can be useful for these and other identities.

50 Use probability identities and Venn diagrams to help answer the following questions.

51 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other

52 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other What s the probability that a student has a known hair color?

53 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other What s the probability that a student has a known hair color? 0.82.

54 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other What s the probability that a student has a known hair color? What s the probability that a student has either red or brown hair?

55 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other What s the probability that a student has a known hair color? What s the probability that a student has either red or brown hair? 0.36.

56 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other What s the probability that a student has a known hair color? What s the probability that a student has either red or brown hair? What s the probability that a student has neither black nor blond hair?

57 Use probability identities and Venn diagrams to help answer the following questions. Example (hair color) The table gives the proportion of hair colors in a class. Black Blond Brown Red White/Grey Other What s the probability that a student has a known hair color? What s the probability that a student has either red or brown hair? What s the probability that a student has neither black nor blond hair? 0.58.

58 When a sample space has a finite possible number of outcomes:

59 When a sample space has a finite possible number of outcomes: Assign a probability to each outcome using some rule/description of the phenomenon. These probabilities must sum to 1 and be between 0 and 1.

60 When a sample space has a finite possible number of outcomes: Assign a probability to each outcome using some rule/description of the phenomenon. These probabilities must sum to 1 and be between 0 and 1. The probability of an event can be calculated as the sum of the probabilities of the outcomes which make it up.

61 When a sample space has a finite possible number of outcomes: Assign a probability to each outcome using some rule/description of the phenomenon. These probabilities must sum to 1 and be between 0 and 1. The probability of an event can be calculated as the sum of the probabilities of the outcomes which make it up. Example (rolling dice) Create the probability model for rolling two dice.

62 When a sample space has a finite possible number of outcomes: Assign a probability to each outcome using some rule/description of the phenomenon. These probabilities must sum to 1 and be between 0 and 1. The probability of an event can be calculated as the sum of the probabilities of the outcomes which make it up. Example (rolling dice) Create the probability model for rolling two dice

63 To construct the previous probability model for two dice, we used the following rule.

64 To construct the previous probability model for two dice, we used the following rule. Theorem (equally likely outcomes) If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1 k. The probability of any event A is P(A) = count of outcomes in A k

65 To construct the previous probability model for two dice, we used the following rule. Theorem (equally likely outcomes) If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1 k. The probability of any event A is P(A) = count of outcomes in A k We didn t use this rule exactly because we pinched together certain outcomes since their result was the same in some sense, but the idea is the same.

66 When a phenomenon includes two procedures like our rolling of two dice, it can be difficult to assign probabilities to the sample space. However, this process simplifies when you can assume that the events are independent.

67 When a phenomenon includes two procedures like our rolling of two dice, it can be difficult to assign probabilities to the sample space. However, this process simplifies when you can assume that the events are independent. Definition (independence) Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, then P(A and B) = P(A) P(B). We call this the multiplication rule for independent events.

68 Example (independent events) Are the following independent events or not?

69 Example (independent events) Are the following independent events or not? 1 Drawing an ace from one deck of cards and then drawing a king from another deck.

70 Example (independent events) Are the following independent events or not? 1 Drawing an ace from one deck of cards and then drawing a king from another deck. 2 Drawing an ace from one deck of cards and then drawing a king from the same deck without replacing the ace first.

71 Example (independent events) Are the following independent events or not? 1 Drawing an ace from one deck of cards and then drawing a king from another deck. 2 Drawing an ace from one deck of cards and then drawing a king from the same deck without replacing the ace first. 3 Event A is disjoint from Event B. Are they independent?

72 With all these rules at our disposal, we can now calculate probabilities of somewhat complicated events.

73 With all these rules at our disposal, we can now calculate probabilities of somewhat complicated events. What s the probability that in a room of 25 people at least two people share a birthday?

74 With all these rules at our disposal, we can now calculate probabilities of somewhat complicated events. What s the probability that in a room of 25 people at least two people share a birthday? 56.87%.

### 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

### Probability Models. Section 6.2

Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example

### 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

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

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

### Lecture 6 Probability

Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times

### 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

### Basic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes.

Basic Probability Ideas Experiment - a situation involving chance or probability that leads to results called outcomes. Random Experiment the process of observing the outcome of a chance event Simulation

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

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

### 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

### 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

### Chapter 1. Probability

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

### Chapter 5 - Elementary Probability Theory

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

### Week 3 Classical Probability, Part I

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

### 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.

### 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

### 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

### 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

### Georgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 6

How Odd? Standards Addressed in this Task MGSE9-12.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not). MGSE9-12.S.CP.7

### 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

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes

### 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

### 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. 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.

### CHAPTERS 14 & 15 PROBABILITY STAT 203

CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical

### 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

### 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.

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

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

### Probability I Sample spaces, outcomes, and events.

Probability I Sample spaces, outcomes, and events. When we perform an experiment, the result is called the outcome. The set of possible outcomes is the sample space and any subset of the sample space is

### 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

### Def: The intersection of A and B is the set of all elements common to both set A and set B

Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:

### 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

### 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

### 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.

### 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

### 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

### 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

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce

### 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

### 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!

### Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules

+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that

### Probability. Sometimes we know that an event cannot happen, for example, we cannot fly to the sun. We say the event is impossible

Probability Sometimes we know that an event cannot happen, for example, we cannot fly to the sun. We say the event is impossible Impossible In summer, it doesn t rain much in Cape Town, so on a chosen

### Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)

12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the

### CSC/MATA67 Tutorial, Week 12

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

### If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

### ECON 214 Elements of Statistics for Economists

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

### 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

### Counting and Probability

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

### 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

### Beginnings of Probability I

Beginnings of Probability I Despite the fact that humans have played games of chance forever (so to speak), it is only in the 17 th century that two mathematicians, Pierre Fermat and Blaise Pascal, set

### Probability. Probabilty Impossibe Unlikely Equally Likely Likely Certain

PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0

### 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

### 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

### CS 361: Probability & Statistics

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

### Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

### 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

### Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

### Math 147 Lecture Notes: Lecture 21

Math 147 Lecture Notes: Lecture 21 Walter Carlip March, 2018 The Probability of an Event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of

### "Well, statistically speaking, you are for more likely to have an accident at an intersection, so I just make sure that I spend less time there.

6.2 Probability Models There was a statistician who, when driving his car, would always accelerate hard before coming to an intersection, whiz straight through it, and slow down again once he was beyond

### STAT 3743: Probability and Statistics

STAT 3743: Probability and Statistics G. Jay Kerns, Youngstown State University Fall 2010 Probability Random experiment: outcome not known in advance Sample space: set of all possible outcomes (S) Probability

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

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

### Module 4 Project Maths Development Team Draft (Version 2)

5 Week Modular Course in Statistics & Probability Strand 1 Module 4 Set Theory and Probability It is often said that the three basic rules of probability are: 1. Draw a picture 2. Draw a picture 3. Draw

### Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules

### Exam III Review Problems

c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous Week-in-Reviews

### 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

### 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

### Probability - Grade 10 *

OpenStax-CNX module: m32623 1 Probability - Grade 10 * Rory Adams Free High School Science Texts Project Sarah Blyth Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative

### Section 6.5 Conditional Probability

Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability

### Objective: Determine empirical probability based on specific sample data. (AA21)

Do Now: What is an experiment? List some experiments. What types of things does one take a "chance" on? Mar 1 3:33 PM Date: Probability - Empirical - By Experiment Objective: Determine empirical probability

### 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

### 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

### What Do You Expect? Concepts

Important Concepts What Do You Expect? Concepts Examples Probability A number from 0 to 1 that describes the likelihood that an event will occur. Theoretical Probability A probability obtained by analyzing

### 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

### [Independent Probability, Conditional Probability, Tree Diagrams]

Name: Year 1 Review 11-9 Topic: Probability Day 2 Use your formula booklet! Page 5 Lesson 11-8: Probability Day 1 [Independent Probability, Conditional Probability, Tree Diagrams] Read and Highlight Station

### PROBABILITY Case of cards

WORKSHEET NO--1 PROBABILITY Case of cards WORKSHEET NO--2 Case of two die Case of coins WORKSHEET NO--3 1) Fill in the blanks: A. The probability of an impossible event is B. The probability of a sure

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

QuestionofDay Question of the Day There are 31 educators from the state of Nebraska currently enrolled in Experimentation, Conjecture, and Reasoning. What is the probability that two participants in our

### 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 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

### 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?

### heads 1/2 1/6 roll a die sum on 2 dice 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 1, 2, 3, 4, 5, 6 heads tails 3/36 = 1/12 toss a coin trial: an occurrence

trial: an occurrence roll a die toss a coin sum on 2 dice sample space: all the things that could happen in each trial 1, 2, 3, 4, 5, 6 heads tails 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 example of an outcome:

### Textbook: pp Chapter 2: Probability Concepts and Applications

1 Textbook: pp. 39-80 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.

### Midterm 2 Practice Problems

Midterm 2 Practice Problems May 13, 2012 Note that these questions are not intended to form a practice exam. They don t necessarily cover all of the material, or weight the material as I would. They are

### 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

### Probability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( )

Probability Test Review Math 2 Name 1. Use the following venn diagram to answer the question: Event A: Odd Numbers Event B: Numbers greater than 10 a. What is? b. What is? c. ( ) d. ( ) 2. In Jason's homeroom

### 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

### 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

### Probability - Chapter 4

Probability - Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person

### 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,

### 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

### 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

### 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

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

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

### Introduction to Probability and Statistics I Lecture 7 and 8

Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes

### CS 361: Probability & Statistics

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

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

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

### 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

### Probability and Counting Rules. Chapter 3

Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of

### 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