# If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics

Size: px
Start display at page:

Download "If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics"

Transcription

1 If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements (Section 5.2) Consider rolling two dice. How likely is it that you get two even numbers OR a sum of 5? What about two even numbers OR a sum of 6? How would you find these probabilities? Let s investigate these and other probabilities. What does OR mean? This may seem obvious but we use OR in math a little differently than in English. When we talk about event E OR event F, we mean either E or F occurs, or possibly both, occur. expl 1: Consider rolling two distinguishable, fair, six-sided dice. The sample space is given below. Answer the following questions. This will help us understand an important formula we will see later. Recall the probability of an event is the number of successes divided by the number of possibilities. a.) What is the probability of getting a sum of 5? Circle the successes above. b.) What is the probability of getting a sum of 7? Circle the successes above. c.) What is the probability of getting a sum of 5 OR a sum of 7? Notice how you have already circled all of the successes. d.) Can the events sum of 5 and sum of 7 happen at the same time (in one roll of the dice)? 1

2 Definition: Two events are disjoint if they have no outcomes in common. This means that they cannot occur at the same time. These are often called mutually exclusive events. This is true of the events sum of 5 and sum of 7. We have a very important rule that applies to mutually exclusive events, which we saw at work in example 1. Addition Rule for Disjoint Events: If E and F are disjoint (or mutually exclusive) events, then PEorF ( ) = PE ( ) + PF ( ). Again, why are sum of 5 and sum of 7 considered disjoint? You may see this written as PE ( F). This is the union symbol. expl 2: We could use set notation and a Venn diagram to help illustrate how these two events cannot coincide. a.) List out, in set notation, the outcomes in the event sum of 5. We ll call this event E. Write the dice results in ordered pair notation. For instance, let (1, 4) represent a 1 on the first die and a 4 on the second die. b.) List out, in set notation, the outcomes in the event sum of 7. We ll call this event F. Write the dice results in ordered pair notation. c.) Now we draw a Venn diagram, with a rectangle encompassing two non-intersecting circles. One circle represents event E and the other represents event F. Place within each circle those outcomes that are within each event. Which outcomes are in neither circle? Where would you place them? You should have no outcomes that are in both circles. 2

3 We can see how this rule may be generalized for any number of mutually exclusive events. Addition Rule for Many Disjoint Events: If E, F, G, H, are disjoint (or mutually exclusive) events, then P( E or F or G or H or...) = P( E) + P( F) + P( G) + P( H ) Would this be true for P(sum of 5) and P(sum of 7) and P(sum of 9) and P(sum of 11)? But what do we do to find probabilities of events that are not mutually exclusive? Let s consider this new example. expl 3: Again, consider the two dice as before. Answer the following questions. a.) What is the probability of getting two even dice? Circle the successes above. b.) What is the probability of getting a sum of 6? Circle the successes above. c.) What is the probability of getting two even dice OR a sum of 6? Can you just add the probabilities from parts a and b as we did before? 3

4 expl 3 continued: d.) The probability for part c can be calculated by counting the number of successes (or, those pairs of dice where the sum is 6 OR the two dice are even) and dividing that by the number of possibilities (which is still 36). Explain why this should be 12/36. Remember, number of successes divided by number of possibilities. e.) If we simply add P(sum is 6) + P(two even dice), we do not get 12/36. What do we get and why is this not equal to the probability we are after? In other words, what are we counting that we should not be? f.) To further clarify, which outcomes satisfy sum is 6 AND two even dice? In other words, find those outcomes that satisfy both of the events. Calculate P(sum is 6 AND two even dice). AND means both events occur. g.) Now draw a Venn diagram, with a rectangle encompassing two intersecting circles. Label one circle sum is 6 and the other two even dice. Can you picture the outcomes in each circle? (You do not need to write them in the circles but you can if you want.) Which outcomes are in the intersection of the two circles? 4

5 This leads us to a very important rule. You may see this written as PE ( F). This is the intersection symbol. The General Addition Rule: For any two events E and F, we know P( E or F) = P( E) + P( F) P( E and F). Since counting the possibilities and successes is a large part of calculating probabilities, you will also see these formulas in terms of the number of outcomes in events and not their probabilities. This can work for any formula in this section, but I show it here only for the General Addition Rule. For instance, you may see N( E or F) = N( E) + N( F) N( E and F), where N(E) means the number of outcomes in event E, etc. Same idea, but uses number of outcomes instead of probabilities. Worksheet: Probability: Addition Rule: This worksheet gives us a quick example of the General Addition Rule. We will use experimental (or empirical) probability to explore the rule. Worksheet: Counting problems involving OR : This worksheet explores counting the successes for probability questions involving OR. We will look at when the events do have outcomes in common and when they do not. Since we can often use the fundamental definition for probability (number of successes divided by number of possibilities), being able to count the successes is vitally important. 5

6 Contingency Tables: Definition: A contingency table or two-way table shows two categories of data and how they interact. Surveys, whose respondents are broken into categories, often result in two-way tables. There will be a row variable and a column variable. Each number is said to be in a cell in the table. expl 4: Below is a contingency table that shows the relationship between cigar smoking and dying from cancer. Answer the questions that follow. Died from Did Not Die Cancer from Cancer Never smoked cigars ,747 Total Former cigar smoker 91 7,757 Current cigar smoker 141 7,725 Total (source: National study of 137,243 U.S. men, Journal of the National Cancer Institute, Feb. 16, 2000) a.) Find and record the totals for each column and each row. This is the first thing you should do with such a table. b.) If a survey respondent was chosen at random, what is the probability that he was a former cigar smoker? c.) If a survey respondent was chosen at random, what is the probability that he died from cancer? d.) If a survey respondent was chosen at random, what is the probability that he was a former cigar smoker OR died from cancer? 6 What is considered a success? How many successes are there? How many possibilities?

7 expl 4 continued: e.) If a survey respondent was chosen at random, what is the probability that he was a former cigar smoker AND died from cancer? f.) If a survey respondent was chosen at random, what is the probability that he was not a former cigar smoker? Definition: Complement: Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E C, is all outcomes in the sample space S that are not outcomes in the event E. The events former cigar smoker and not a former cigar smoker are complements. Consider rolling one six-sided die. Can you think of two events that are complements? There are other notations used for the complement, like E, E, or E. expl 5: I have a bag with ten marbles: four red, three yellow, and three blue. a.) If I select one marble from the bag, what is the probability that it is blue? b.) If I select one marble from the bag, what is the probability that it is not blue? 7 How many are not blue?

8 expl 6: I will select a card from the face cards from a poker deck. (The face cards are Jack of Hearts, Queen of Hearts, King of Hearts, Jack of Diamonds, Queen of Diamonds, King of Diamonds, Jack of Spades, Queen of Spades, King of Spades, Jack of Clubs, Queen of Clubs, and King of Clubs. Clubs and Spades are black, Hearts and Diamonds are red.) a.) If I select one card from my partial deck, what is the probability that it is a black Queen? b.) If I select one card from my partial deck, what is the probability that it is not a black Queen? How many cards are not black Queens? These probabilities involving complements can be done the traditional method of counting the successes and possibilities. However, we could also use a basic rule. Complement Rule: If E represents any event and E C represents the complement of E, then P(E C ) = 1 P(E). Since E and E C make up all possibilities, their probabilities must add to 1. Redo example 5 but use this formula. (This assumes you did not use the formula when you did the problem originally.) expl 5 again: I have a bag with ten marbles: four red, three yellow, and three blue. a.) If I select one marble from the bag, what is the probability that it is blue? b.) If I select one marble from the bag, what is the probability that it is not blue? Use the formula. 8

### 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

### 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

### 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

### 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

### 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. 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

### 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 Rules. 2) The probability, P, of any event ranges from which of the following?

Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,

### 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 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,

### 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

### Mutually Exclusive Events

Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number and you roll a 2? Can these both occur at the same time? Why or why not? Mutually

### 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

### Applications of Probability

Applications of Probability CK-12 Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive

### 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:

### 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

### Classical vs. Empirical Probability Activity

Name: Date: Hour : Classical vs. Empirical Probability Activity (100 Formative Points) For this activity, you will be taking part in 5 different probability experiments: Rolling dice, drawing cards, drawing

### More Probability: Poker Hands and some issues in Counting

More Probability: Poker Hands and some issues in Counting Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the

### (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

### 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 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

### 2.5 Sample Spaces Having Equally Likely Outcomes

Sample Spaces Having Equally Likely Outcomes 3 Sample Spaces Having Equally Likely Outcomes Recall that we had a simple example (fair dice) before on equally-likely sample spaces Since they will appear

### Venn Diagram Problems

Venn Diagram Problems 1. In a mums & toddlers group, 15 mums have a daughter, 12 mums have a son. a) Julia says 15 + 12 = 27 so there must be 27 mums altogether. Explain why she could be wrong: b) There

### 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:

### 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.

### 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. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College

Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical

### 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,

### Objective 1: Simple Probability

Objective : Simple Probability To find the probability of event E, P(E) number of ways event E can occur total number of outcomes in sample space Example : In a pet store, there are 5 puppies, 22 kittens,

### Math 227 Elementary Statistics. Bluman 5 th edition

Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical

### Developed by Rashmi Kathuria. She can be reached at

Developed by Rashmi Kathuria. She can be reached at . Photocopiable Activity 1: Step by step Topic Nature of task Content coverage Learning objectives Task Duration Arithmetic

### Unit 11 Probability. Round 1 Round 2 Round 3 Round 4

Study Notes 11.1 Intro to Probability Unit 11 Probability Many events can t be predicted with total certainty. The best thing we can do is say how likely they are to happen, using the idea of probability.

### 4. Are events C and D independent? Verify your answer with a calculation.

Honors Math 2 More Conditional Probability Name: Date: 1. A standard deck of cards has 52 cards: 26 Red cards, 26 black cards 4 suits: Hearts (red), Diamonds (red), Clubs (black), Spades (black); 13 of

### Day 5: Mutually Exclusive and Inclusive Events. Honors Math 2 Unit 6: Probability

Day 5: Mutually Exclusive and Inclusive Events Honors Math 2 Unit 6: Probability Warm-up on Notebook paper (NOT in notes) 1. A local restaurant is offering taco specials. You can choose 1, 2 or 3 tacos

### 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

### Algebra II Probability and Statistics

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Slide 3 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional Probability

### 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

### 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.

### Algebra II. Sets. Slide 1 / 241 Slide 2 / 241. Slide 4 / 241. Slide 3 / 241. Slide 6 / 241. Slide 5 / 241. Probability and Statistics

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Slide 3 / 241 Slide 4 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional

### Lesson 3 Dependent and Independent Events

Lesson 3 Dependent and Independent Events When working with 2 separate events, we must first consider if the first event affects the second event. Situation 1 Situation 2 Drawing two cards from a deck

### GSE Honors Geometry. 1. Create a lattice diagram representing the possible outcomes for the two tiles

GSE Honors Geometry Unit 9 Applications of Probability Name Unit Test Review Part 1 You and a friend have made up a game that involves drawing one numbered tile out of each of two separate bags. The first

### CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY

CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many real-world fields, such as insurance, medical research, law enforcement, and political science. Objectives:

### 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

### Exam 2 Review F09 O Brien. Finite Mathematics Exam 2 Review

Finite Mathematics Exam Review Approximately 5 0% of the questions on Exam will come from Chapters, 4, and 5. The remaining 70 75% will come from Chapter 7. To help you prepare for the first part of the

### 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

### Algebra II. Slide 1 / 241. Slide 2 / 241. Slide 3 / 241. Probability and Statistics. Table of Contents click on the topic to go to that section

Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 2016-01-15 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 241 Sets Independence and Conditional Probability

### Introduction to probability

Introduction to probability Suppose an experiment has a finite set X = {x 1,x 2,...,x n } of n possible outcomes. Each time the experiment is performed exactly one on the n outcomes happens. Assign each

### 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.

### 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

### 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

### Fall (b) Find the event, E, that a number less than 3 is rolled. (c) Find the event, F, that a green marble is selected.

Fall 2018 Math 140 Week-in-Review #6 Exam 2 Review courtesy: Kendra Kilmer (covering Sections 3.1-3.4, 4.1-4.4) (Please note that this review is not all inclusive) 1. An experiment consists of rolling

### Name: Exam 1. September 14, 2017

Department of Mathematics University of Notre Dame Math 10120 Finite Math Fall 2017 Name: Instructors: Basit & Migliore Exam 1 September 14, 2017 This exam is in two parts on 9 pages and contains 14 problems

### 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

### 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.

### PROBABILITY. Example 1 The probability of choosing a heart from a deck of cards is given by

Classical Definition of Probability PROBABILITY Probability is the measure of how likely an event is. An experiment is a situation involving chance or probability that leads to results called outcomes.

### 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

### 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

### 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

### Statistics Intermediate Probability

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

### 7 5 Compound Events. March 23, Alg2 7.5B Notes on Monday.notebook

7 5 Compound Events At a juice bottling factory, quality control technicians randomly select bottles and mark them pass or fail. The manager randomly selects the results of 50 tests and organizes the data

### 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

### [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

### 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

### 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

### 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

### Probability is the likelihood that an event will occur.

Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main

### Intermediate Math Circles November 1, 2017 Probability I. Problem Set Solutions

Intermediate Math Circles November 1, 2017 Probability I Problem Set Solutions 1. Suppose we draw one card from a well-shuffled deck. Let A be the event that we get a spade, and B be the event we get an

### 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

### n(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)

The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability

### Chapter 4 Student Lecture Notes 4-1

Chapter 4 Student Lecture Notes 4-1 Basic Business Statistics (9 th Edition) Chapter 4 Basic Probability 2004 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic Probability Concepts Sample spaces and events,

### 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.

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

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

### Slide 1 Math 1520, Lecture 13

Slide 1 Math 1520, Lecture 13 In chapter 7, we discuss background leading up to probability. Probability is one of the most commonly used pieces of mathematics in the world. Understanding the basic concepts

### 1) If P(E) is the probability that an event will occur, then which of the following is true? (1) 0 P(E) 1 (3) 0 P(E) 1 (2) 0 P(E) 1 (4) 0 P(E) 1

Algebra 2 Review for Unit 14 Test Name: 1) If P(E) is the probability that an event will occur, then which of the following is true? (1) 0 P(E) 1 (3) 0 P(E) 1 (2) 0 P(E) 1 (4) 0 P(E) 1 2) From a standard

### 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

### ABC High School, Kathmandu, Nepal. Topic : Probability

BC High School, athmandu, Nepal Topic : Probability Grade 0 Teacher: Shyam Prasad charya. Objective of the Module: t the end of this lesson, students will be able to define and say formula of. define Mutually

### 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

### UNIT 4 APPLICATIONS OF PROBABILITY Lesson 1: Events. Instruction. Guided Practice Example 1

Guided Practice Example 1 Bobbi tosses a coin 3 times. What is the probability that she gets exactly 2 heads? Write your answer as a fraction, as a decimal, and as a percent. Sample space = {HHH, HHT,

### 4.3 Finding Probability Using Sets

4.3 Finding Probability Using ets When rolling a die with sides numbered from 1 to 20, if event A is the event that a number divisible by 5 is rolled: a) What is the sample space,? b) What is the event

### 0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.

1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. d. a. Copy the table and add a column to show the experimental probability of the spinner landing on

### 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

### 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!

### 5 Elementary Probability Theory

5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one

### 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

### MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability)

MATHEMATICS 152, FALL 2004 METHODS OF DISCRETE MATHEMATICS Outline #10 (Sets and Probability) Last modified: November 10, 2004 This follows very closely Apostol, Chapter 13, the course pack. Attachments

### Basic Concepts of Probability and Counting Section 3.1

Basic Concepts of Probability and Counting Section 3.1 Summer 2013 - Math 1040 June 17 (1040) M 1040-3.1 June 17 1 / 12 Roadmap Basic Concepts of Probability and Counting Pages 128-137 Counting events,

### EXAM. Exam #1. Math 3371 First Summer Session June 12, 2001 ANSWERS

EXAM Exam #1 Math 3371 First Summer Session 2001 June 12, 2001 ANSWERS i Give answers that are dollar amounts rounded to the nearest cent. Here are some possibly useful formulas: A = P (1 + rt), A = P

### Name Date. Goal: Understand sets and set notation.

F Math 12 3.1 Types of Sets and Set Notation p. 146 Name Date Goal: Understand sets and set notation. 1. set: A collection of distinguishable objects; for example, the set of whole numbers is W = {0, 1,

### Test 2 Review Solutions

Test Review Solutions. A family has three children. Using b to stand for and g to stand for, and using ordered triples such as bbg, find the following. a. draw a tree diagram to determine the sample space

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

Statistics Homework Ch 5 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

### CMPSCI 240: Reasoning Under Uncertainty First Midterm Exam

CMPSCI 240: Reasoning Under Uncertainty First Midterm Exam February 18, 2015. Name: ID: Instructions: Answer the questions directly on the exam pages. Show all your work for each question. Providing more

### 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

### Probability: introduction

May 6, 2009 Probability: introduction page 1 Probability: introduction Probability is the part of mathematics that deals with the chance or the likelihood that things will happen The probability of an

### 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