7.5: Conditional Probability

Size: px
Start display at page:

Download "7.5: Conditional Probability"

Transcription

1 7.5: Conditional Probability Example 1: A survey is done of people making purchases at a gas station: buy drink (D) no drink (Dc) Total Buy drink(d) No drink(d c ) Total Buy Gas (G) No Gas (G c ) Total a) What is the probability that a person buys a drink? b) What is the probability that a person doesnt buy a drink? c) What is the probability that a person buys gas and a drink? d) What is the probability that a person buys gas but not a drink? e) What is the probability that a person who buys a drink also buys gas? f) What is the probability that a person who doesnt buy a drink buys gas? 1

2 Definition: If E and F are events in an experiment and P (E) 0, then the conditional probability that the event F will occur given that the event E has already occurred is P (F E) = P (E F ). P (E) Definition: The Product Rule is found by rearranging the above formula as follows: P (E F ) = P (E)P (F E). 2

3 Example 2: Lets use a tree diagram to help us understand the product rule: Example 3: At a party, 1 3 of the guest are women. Seventy-five percent of the women wore sandals and 40 % of the men wore sandals. a) What is the probability that a person chosen at random at the party is a man wearing sandals? b) What is the probability that a person chosen at random is wearing sandals? 3

4 Example 4: Consider drawing 3 cards from a standard deck of 52 cards without replacement. a) What is the probability that the 3 cards are hearts? b) What is the probability that the third card drawn is a heart given the first two cards are hearts? 4

5 Independent events Definition: If A and B are independent events, then P (A) = P (A B) and P (B) = P (B A). Thus, two events A and B are independent if and only if P (A B) = P (A) P (B). Example 5: A medical experiment showed the probability that a new medicine was effective was.75, the probability of a certain side effect was.4, and the probability of both occuring was.3. Are the events independent? 5

6 Example 6: Meghan and Natalie go to Freebirds. After ordering their food they get to roll a pair of fair dice for a chance to get their meal for free. Each die has six sides with one of the sides having a backwards F on it. If both of the dice land on the backwards F, you win. What is the probability that at least one of the girls wins? 6

7 7.7 Bayes Theorem Example 7: If we are given information about P (F E), can we find P (E F )? Definition: The above formula is known as Bayes Theorem. 7

8 Example 8: We are to choose a marble from a cup or a bowl. We flip a fair coin to decide whether to choose from the cup or the bowl. The bowl contains 1 red and 2 green marbles. The cup contains 3 red and 2 green marbles. What is the probability that a marble came from the bowl given that it is red? Example 9: A crate contains 7 basketballs and 4 footballs. A bag contains 4 basketballs and 2 footballs. A ball is drawn at random from the crate and put in the bag. A ball is then drawn from the bag. Given that a basketball was chosen from the bag, what is the probability that a football was drawn from the crate? 8

9 Example 10: Two cards are drawn in succession without replacement from a standard deck of 52 cards. What is the probability that the first card is a face card given that the second card is an ace? Example 11:Complete the following tree diagram and use it to answer the following questions: a) Find P (E). b) Find P (A D). c) Find P (B E). d) Are E and B independent events? 9

10 8.1:Random Variables and Histograms Definition: A random variable is a rule that assigns a number to each outcome of an experiment. Example 1: Suppose we toss a coin three times. Then we could define the random variable X to represent the number of times we get tails. Example 2: Suppose we roll a die until a 5 is facing up. Then we could define the random variable Y to represent the number of times we rolled the die. Example 3: Suppose a flashlight is left on until the battery runs out. Then we could define the random variable Z to represent the amount of time that passed. 10

11 Types of Random Variables 1. A finite discrete random variable is one which can only take on a limited number of values that can be listed. 2. A infinite discrete random variable is one which can take on an unlimited number of values that can be listed in some sort of sequence. 3. A continuous random variable is one which takes on any of the infinite number of values in some interval of real numbers. (i.e. usually measurements) 11

12 Example 4: Classify each of the following random variables as finite discrete, infinite discrete, or continuous and list/describe the possible values for the random variable. a) A drawer contains 15 red pens, 8 blue pens, and 5 black pens. An experiment consists of drawing pens out of the drawer with replacement until a red pen is drawn. Let X represent the number of pens drawn. b) An experiment consists of randomly selecting 3 pens without replacement from a conveyor belt. Let Y represent the total weight (in ounces) of the selected pens. c) An experiment consists of randomly selecting 3 pens with replacement from a drawer that contains 15 red pens, 8 blue pens, and 5 black pens. Let Z represent the number of red pens drawn. 12

13 Example 5: Referring to Example 1, find the probability distribution of the random variable X. Example 6: An experiment consists of randomly selecting a sample of 3 grapes out of a bowl that contains 20, of which 8 are rotten. Let X represent the number of rotten grapes in the sample. Find the probability distribution of X. 13

14 Example 7: Blue Baker prides itself on having the best chocolate chips cookies in town. To be sure each cookie has the right number of chocolate chips, a few cookies are selected from each batch and the number of chocolate chips in each cookie is counted. This is done for several days and the following results were found: Number of Cookies Number of Chocolate Chips Identify the random variable X in this experiment and find the probability distribution of X. 14

15 We use histograms to represent the probability distributions of random variables. We place the possible values for the random variable X on the horizontal axis. We then center a bar around each x value and let its height be equal to the probability of that x value Example 8: Referring to Example 7, a) Draw the histogram for the random variable X b) Find P (X 12) 15

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

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..

More information

Section 6.2 Definition of Probability

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

More information

Math 3C Homework 3 Solutions

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

More information

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

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

More information

Topic : Probability of a Complement of an Event- Worksheet 1. Do the following:

Topic : Probability of a Complement of an Event- Worksheet 1. Do the following: Topic : Probability of a Complement of an Event- Worksheet 1 1. You roll a die. What is the probability that 2 will not appear 2. Two 6-sided dice are rolled. What is the 3. Ray and Shan are playing football.

More information

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

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

More information

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? 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

More information

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

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

More information

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

More information

Find the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.

Find the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd. Math 0 Practice Test 3 Fall 2009 Covers 7.5, 8.-8.3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. ) If a single

More information

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 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

More information

Hoover High School Math League. Counting and Probability

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

More information

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

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum

More information

Statistics 100A Homework 2 Solutions

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

More information

Chapter 4 - Practice Problems 1

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

More information

Section 6-5 Sample Spaces and Probability

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)

More information

Probabilistic Strategies: Solutions

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

More information

2. Three dice are tossed. Find the probability of a) a sum of 4; or b) a sum greater than 4 (may use complement)

2. Three dice are tossed. Find the probability of a) a sum of 4; or b) a sum greater than 4 (may use complement) Probability Homework Section P4 1. A two-person committee is chosen at random from a group of four men and three women. Find the probability that the committee contains at least one man. 2. Three dice

More information

Determine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.

Determine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard. Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5

More information

Definition and Calculus of Probability

Definition and Calculus of Probability In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

More information

AMS 5 CHANCE VARIABILITY

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

More information

Chapter 5 A Survey of Probability Concepts

Chapter 5 A Survey of Probability Concepts Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible

More information

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

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

More information

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

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

More information

6.3 Conditional Probability and Independence

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

More information

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 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

More information

Sample Term Test 2A. 1. A variable X has a distribution which is described by the density curve shown below:

Sample Term Test 2A. 1. A variable X has a distribution which is described by the density curve shown below: Sample Term Test 2A 1. A variable X has a distribution which is described by the density curve shown below: What proportion of values of X fall between 1 and 6? (A) 0.550 (B) 0.575 (C) 0.600 (D) 0.625

More information

Assessment For The California Mathematics Standards Grade 6

Assessment For The California Mathematics Standards Grade 6 Introduction: Summary of Goals GRADE SIX By the end of grade six, students have mastered the four arithmetic operations with whole numbers, positive fractions, positive decimals, and positive and negative

More information

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 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

More information

36 Odds, Expected Value, and Conditional Probability

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

More information

Chapter 4 - Practice Problems 2

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

More information

AP Stats - Probability Review

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

More information

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 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

More information

Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

More information

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

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

More information

1 Combinations, Permutations, and Elementary Probability

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

More information

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

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

More information

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

More information

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 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

More information

ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

More information

Exam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS

Exam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS Exam Name How many distinguishable permutations of letters are possible in the word? 1) CRITICS 2) GIGGLE An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm,

More information

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. 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

More information

Basic Probability Theory II

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

More information

AP Statistics 7!3! 6!

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!

More information

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0. Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

More information

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

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

More information

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

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.

More information

Chapter 16: law of averages

Chapter 16: law of averages Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

More information

Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2

Math 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2 Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable

More information

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

(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

More information

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

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

More information

Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

More information

Chapter 13 & 14 - Probability PART

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

More information

13.0 Central Limit Theorem

13.0 Central Limit Theorem 13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated

More information

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

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

More information

Betting systems: how not to lose your money gambling

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

More information

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].

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

More information

Formula for Theoretical Probability

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

More information

Name: Date: Use the following to answer questions 2-4:

Name: Date: Use the following to answer questions 2-4: Name: Date: 1. A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event A occurs is recorded. What does this proportion represent? A) The probability

More information

Chapter 5 - Practice Problems 1

Chapter 5 - Practice Problems 1 Chapter 5 - Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level

More information

WHERE DOES THE 10% CONDITION COME FROM?

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

More information

CAHSEE Algebra Cluster #4: Statistics, Data Analysis, an Probability Name: Cluster #4 Review

CAHSEE Algebra Cluster #4: Statistics, Data Analysis, an Probability Name: Cluster #4 Review CAHSEE Algebra Cluster #4: Statistics, Data Analysis, an Probability Name: Cluster #4 Review 1. The number of classic book Nanette sells in her 2. Which scatterplot shows a positive correlation? bookshop

More information

Probability and Venn diagrams UNCORRECTED PAGE PROOFS

Probability and Venn diagrams UNCORRECTED PAGE PROOFS Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve

More information

Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.

Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary. Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate

More information

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

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

More information

PROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1.

PROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1. PROBABILITY SIMPLE PROBABILITY SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and. There are two categories of simple probabilities. THEORETICAL

More information

WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM

WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM EXAMPLE 1. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed.

More information

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

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

More information

Math 202-0 Quizzes Winter 2009

Math 202-0 Quizzes Winter 2009 Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

More information

Probability of Compound Events

Probability of Compound Events Probability of Compound Events Why? Then You calculated simple probability. (Lesson 0-11) Now Find probabilities of independent and dependent events. Find probabilities of mutually exclusive events. Online

More information

Differentiated plans for Years 3 & 4 for Chance & Data

Differentiated plans for Years 3 & 4 for Chance & Data Differentiated plans for Years 3 & 4 for Chance & Data Ian Lowe, MAV Professional Officer, 2008 IF YOU USE ANY OF THIS PLEASE PROVIDE FEEDBACK TO IAN AT ilowe@mav.vic.edu.au THIS WILL QUALIFY YOU FOR AN

More information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

More information

Probability definitions

Probability definitions Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating

More information

Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev

Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland

More information

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

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

More information

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 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

More information

The Procedures of Monte Carlo Simulation (and Resampling)

The Procedures of Monte Carlo Simulation (and Resampling) 154 Resampling: The New Statistics CHAPTER 10 The Procedures of Monte Carlo Simulation (and Resampling) A Definition and General Procedure for Monte Carlo Simulation Summary Until now, the steps to follow

More information

Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations)

Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations) Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations) Note: At my school, there is only room for one math main lesson block in ninth grade. Therefore,

More information

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 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

More information

Homework 8 Solutions

Homework 8 Solutions CSE 21 - Winter 2014 Homework Homework 8 Solutions 1 Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that

More information

Name Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Name Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Review Problems for Mid-Term 1, Fall 2012 (STA-120 Cal.Poly. Pomona) Name Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether

More information

TEACHING ADULTS TO REASON STATISTICALLY

TEACHING ADULTS TO REASON STATISTICALLY TEACHING ADULTS TO REASON STATISTICALLY USING THE LEARNING PROGRESSIONS T H E N AT U R E O F L E A R N I N G Mā te mōhio ka ora: mā te ora ka mōhio Through learning there is life: through life there is

More information

Mathematical goals. Starting points. Materials required. Time needed

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

More information

PROBABILITY AND RELATIVE FREQUENCY

PROBABILITY AND RELATIVE FREQUENCY Edexcel GCSE Mathematics (Linear) 1MA0 PROBABILITY AND RELATIVE FREQUENCY Materials required for examination Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser.

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 0.4987 B) 0.9987 C) 0.0010 D) 0.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 0.4987 B) 0.9987 C) 0.0010 D) 0. Ch. 5 Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 1 Find Areas Under the Standard Normal Curve 1) Find the area under the standard normal

More information

More Multi Step Problems. Miscellaneous Themes

More Multi Step Problems. Miscellaneous Themes More Multi Step Problems Miscellaneous Themes Jelly beans cost $2.00 for 4 packages. There were 25 students. How much money will be needed so that each student can get an equal amount of jellybeans? How

More information

5) The table below describes the smoking habits of a group of asthma sufferers. two way table ( ( cell cell ) (cell cell) (cell cell) )

5) The table below describes the smoking habits of a group of asthma sufferers. two way table ( ( cell cell ) (cell cell) (cell cell) ) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine which score corresponds to the higher relative position. 1) Which score has a better relative

More information

The Binomial Probability Distribution

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

More information

EDEXCEL FUNCTIONAL SKILLS PILOT

EDEXCEL FUNCTIONAL SKILLS PILOT EEXEL FUNTIONAL SKILLS PILOT Maths Level hapter 7 Working with probability SETION K Measuring probability 9 Remember what you have learned 3 raft for Pilot Functional Maths Level hapter 7 Pearson Education

More information

Second Midterm Exam (MATH1070 Spring 2012)

Second Midterm Exam (MATH1070 Spring 2012) Second Midterm Exam (MATH1070 Spring 2012) Instructions: This is a one hour exam. You can use a notecard. Calculators are allowed, but other electronics are prohibited. 1. [60pts] Multiple Choice Problems

More information

Probability and Statistics is one of the strands tested on the California Standards Test.

Probability and Statistics is one of the strands tested on the California Standards Test. Grades 3-4 Probability and Statistics is one of the strands tested on the California Standards Test. Probability is introduced in 3 rd grade. Many students do not work on probability concepts in 5 th grade.

More information

Methods Used for Counting

Methods Used for Counting COUNTING METHODS From our preliminary work in probability, we often found ourselves wondering how many different scenarios there were in a given situation. In the beginning of that chapter, we merely tried

More information

MATH 1108 R07 MIDTERM EXAM 1 SOLUTION

MATH 1108 R07 MIDTERM EXAM 1 SOLUTION MATH 1108 R07 MIDTERM EXAM 1 SOLUTION FALL 2015 - MOON Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. Do not use

More information

You flip a fair coin four times, what is the probability that you obtain three heads.

You flip a fair coin four times, what is the probability that you obtain three heads. Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

More information

ECE-316 Tutorial for the week of June 1-5

ECE-316 Tutorial for the week of June 1-5 ECE-316 Tutorial for the week of June 1-5 Problem 35 Page 176: refer to lecture notes part 2, slides 8, 15 A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same

More information

Tasks to Move Students On

Tasks to Move Students On Maths for Learning Inclusion Tasks to Move Students On Part 1 Maths for Learning Inclusion (M4LI) Tasks to Move Students On Numbers 1 10 Structuring Number Maths for Learning Inclusion (M4LI) Tasks to

More information

PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

More information

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. (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

More information

Chapter 4 Lecture Notes

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,

More information

MATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker)

MATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker) Last modified: February, 00 References: MATHEMATICS 5, SPRING 00 PROBABILITY THEORY Outline # (Combinatorics, bridge, poker) PRP(Probability and Random Processes, by Grimmett and Stirzaker), Section.7.

More information