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

Size: px
Start display at page:

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

Transcription

1 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 theory, D. Reidel Publishing Company, D. Tel shook his head as he finished examining his patient. You have a very serious disease, he said, of ten people who have got this disease only one survives. As the patient was sufficiently scared by this information, D. Tel began to console him: But you are very lucky sir, because you came to me. I have already had nine patients who all died of it, so you will survive. Terminology: n(e)=the number of ways an event can occur with success. n(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. The probability of an event E, n(s) assuming all events are equally likely to occur. Example: Suppose you roll a die. It can land with a 1,2,3,4,5, or 6 showing. n(s)=6. Question: What is the probability of getting a 1 or a 2 showing? A success is if it lands 1 or 2. So n(e)=2, and n(s) = 2 6 = 1 3. Odds: The odds of an event occurring, s(e) = n(e) : n(e ), where n(e )=the number of ways of having a failure and n(e) is the number of ways of having a success.. Example: to find the odds of getting a 1 or 2 with a single roll of a die :

2 n(e) = the number of ways of having a success = 2. n(e )=the number of ways of having a failure. You have a failure if it lands 3,4,5, or 6. So n(e )=4. Therefore s(e) = n(e): n(e ) = 2 : 4 = 1 : 2 What is the probability of getting (a total of) 6 when a pair of dice is rolled? Wrong way: It is possible to get a total of 2,3,4,5,6,7,8,9,10,11, or 12. So the probability of getting a 6 is n(e)/n(s)=1/11 since there is one way to get a sum of 6, hence n(e)=1, and there are 11 possible sums so n(s)=11. The last sentence contains an error. What is the error in the above calculation? Not all ways are getting a sum are equally likely. For example, there is only one way to get a sum of two: Both dice must show one. However, if you consider the two dice as die 1 and die 2, you could have die 1 = 1 and die 2 = 5 or die 1 = 2 and die 2 = 4, etc. We must count the ways so all outcomes are equally likely. There are 6 ways for die 1 to land and 6 ways for die 2 to land, and counting this way there are 36 ways both dice can land. so n(s)=36. Die 1 and die 2 can land as (1,5), (2,4), (3,3), (4,2), (5,1) so n(e)=5. Therefore p(e) = 5/36 A coin is tossed ten times. What is the probability that it lands heads all ten times? There are two ways the coin can land on the first toss {heads, tails }. There are two ways the coin can land on the second toss. So there are 2 2 ways the coin can land on the first two tosses. Continuing, there are = 2 10 = 1024 ways the coin can land in ten tosses. n(e)=the number of ways it can land heads all ten times=1.

3 /n(s) = 1/1024. A random elevator rider is equally likely to exit on one of six floors. Three random riders enter the elevator. What is the probability that they all exit on different floors? Let n(e) be the number of ways that they can exit on different floors. The first rider can exit on any one of 6 floors. After that, the second rider has 5 choices, and after that the third rider has 4 choices. So n(e) = The total number of ways three riders can exit is n(s) = n(s) = = 5 9 Three people are chosen at random. What is the probability that they all have different birthdays? What is the probability that at least two of the three people have the same birthday? Assume a year has 365 days (so we are ignoring the complication of leap years.) n(s)=the number of ways three people can have birthdays = Let n(e)=the number of ways three people can have different birthdays. n(e) = = 365 P 3. So n(s) = Let E be the event that at least two people have the same birthday. We note that n(e) + n(e ) = n(s), and dividing each term by n(s), we see that This formula is true in general. In our example, p(e) + p(e ) = 1. p(e ) =.008

4 Thirty people are chosen at random. What is the probability that they all have different birthdays? What is the probability that at least two of the thirty people have the same birthday? Assume a year has 365 days (so we are ignoring the complication of leap years.) n(s)=the number of ways thirty people can have birthdays = Let n(e)=the number of ways thirty people can have different birthdays. n(e) = 365 P 30. So we have n(s) = 365 P (365) 30 If you use a TI calculator, you would push 365, MATH, PRB, then npr, then 30, ENTER, and then divide by !. When you calculate the numerator 365 P 30 = (365 30)! = 365!, if you attempt to calculate 333! 365! and then divide by 333!, you will get overflow. If you are working near a computer that is running and don t have a calculator handy, the following BASIC code, inserted into a procedure as we have done before (run Excel, click on tools, macro, Visual Basic Editor, insert module, then insert procedure, name the procedure, then insert the code, then run the procedure), will compute n(e): a=1 For i=0 to 29 a=a*(365-i)/365 Next i Cells(1,1)=a Let E be the event that at least two people have the same birthday. Since p(e) + p(e ) = 1, p(e ) =.706 This means that if you run the computer simulation of the experiment 1000 times (see the Friday assignment,) you expect that approximately 706 times at least two people will have the same birthday.

5 We already mentioned that p(e) + p(e ) = 1. More basic rules: E F denotes the event that an event from E or from F (or both) occurs. E F denote the event that an event from E and F occurs. If E and F are mutually exclusive, If E and F are not mutually exclusive, p(e F ) = p(e) + p(f ). p(e F ) = p(e) + p(f ) p(e F ). Example: 1) One card is drawn from a deck. a) What is the probability that it is either a king or a queen? n(e)=8 since there are four kings and four queens. n(s)=52, so /n(s) = 8/52 = 2/13 b) What is the probability that it is neither a king or a queen? If we simply use the above observation, p(e) = 1 p(e ) = 1 2/13 = 11/13 2) Suppose you draw one card from a deck. Let E be the event that the card is red, and F be the event that the card is black. E F represents the event that the card is red or black. E and F are mutually exclusive, and

6 1 = p(e F ) = p(e) + p(f ) = 1/2 + 1/2. 3) Now let E be the event that the card is red, and F be the event that the card is a king or queen. E and F are not mutually exclusive, and p(e F ) is the probability that a card is a red king or queen. n(s)=52 = the number of ways of choosing a card. There are 4 cards that are red and kings or queens, so p(e F ) = 4/52. Also, p(f ) = 8/52 since there are 8 cards in a deck that are kings or queens. E F represents the event that the card is red or a king or queen. The word or is not exclusive. That means if a card is red and a king, for example, then that is an event in E F. p(e F ) = p(e) + p(f ) p(e F ) = 1/2 + 8/52 4/52 = 15/26. In poker (you are dealt five cards), what is the probability of being dealt a royal flush? (an ace, king, queen, jack, and ten all of the same suit). The number of ways of being dealt five cards is 52 C 5. There are 4 ways you can get a royal flush (once for each suit). p(e) = 4/( 52 C 5 ) = 4/ = 1/ shoes are selected at random from a pile containing 8 left shoes and 12 right shoes. What is the probability of ending up with two pairs of shoes? A pair of shoes will mean any left shoe and any right shoe. You must choose 2 shoes from the 8 left shoes and 2 shoes from the 12 right shoes. The number of ways of doing this is n(e) = 8 C 2 12 C 2 = 1848 The total number of ways of choosing 4 shoes from the pile of 20 shoes is 20 C 4 = 4845

7 Then the probability /n(s) = 1848/4845 = 616/1615 In a five card hand, what is the probability of getting exactly 4 cards of the same suit? n(e) is the number of ways of choosing a suit (4) and then choosing exactly 4 cards from that suit, and then choosing the last card from the rest of the deck. Having chosen a suit, there are 13 C 4 ways of choosing four cards from that suit. The remaining card must be chosen from the 39 other cards (of a different suit). Therefore n(e) = 4 13 C The total number of ways of choosing 5 cards from the deck is 52 C 5. n(s) = 4 13C C 5 = = In a five card hand, what is the probability of getting exactly 4 of a kind, e.g, four aces,..., or four kings? There are 13 ways of choosing which kind: ace, two, three,...,king. Having made that choice, there are 48 ways to choose the fifth card. So n(e) = The total number of ways of choosing 5 cards from the deck is 52 C 5. n(s) = C 5 = End of Lecture

### 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

More information

### 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

More information

### 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

More information

### 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.

More information

### 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

More information

### 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:

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### 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.

More information

### 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

More information

### 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,

More information

### Probability and Statistics. Copyright Cengage Learning. All rights reserved.

Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.2 Probability Copyright Cengage Learning. All rights reserved. Objectives What Is Probability? Calculating Probability by

More information

### 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,

More information

### 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,

More information

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

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

More information

### 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

More information

### 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

More information

### 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:

More information

### LISTING THE WAYS. getting a total of 7 spots? possible ways for 2 dice to fall: then you win. But if you roll. 1 q 1 w 1 e 1 r 1 t 1 y

LISTING THE WAYS A pair of dice are to be thrown getting a total of 7 spots? There are What is the chance of possible ways for 2 dice to fall: 1 q 1 w 1 e 1 r 1 t 1 y 2 q 2 w 2 e 2 r 2 t 2 y 3 q 3 w 3

More information

### 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

More information

### 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

More information

### STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes

STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle

More information

### Mathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015

1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:

More information

### 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

More information

### 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.

More information

### Independent Events. 1. Given that the second baby is a girl, what is the. e.g. 2 The probability of bearing a boy baby is 2

Independent Events 7. Introduction Consider the following examples e.g. E throw a die twice A first thrown is "" second thrown is "" o find P( A) Solution: Since the occurrence of Udoes not dependu on

More information

### Math 14 Lecture Notes Ch. 3.3

3.3 Two Basic Rules of Probability If we want to know the probability of drawing a 2 on the first card and a 3 on the 2 nd card from a standard 52-card deck, the diagram would be very large and tedious

More information

### 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

More information

### 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

More information

### Math 4610, Problems to be Worked in Class

Math 4610, Problems to be Worked in Class Bring this handout to class always! You will need it. If you wish to use an expanded version of this handout with space to write solutions, you can download one

More information

### 1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,

More information

### 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

More information

### 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

More information

### 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 information

### Unit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22

Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/2011 1 / 22 Chapter 13: What is a probability Denition The probability that an event happens is the percentage

More information

### 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

More information

### Unit 6: Probability. Marius Ionescu 10/06/2011. Marius Ionescu () Unit 6: Probability 10/06/ / 22

Unit 6: Probability Marius Ionescu 10/06/2011 Marius Ionescu () Unit 6: Probability 10/06/2011 1 / 22 Chapter 13: What is a probability Denition The probability that an event happens is the percentage

More information

### 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

More information

### 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,

More information

### 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

More information

### 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

More information

### 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

More information

### 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

More information

### , -the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.

4-1 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,

More information

### 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

More information

### 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

More information

### 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.

More information

### 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.

More information

### 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

More information

### Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

More information

### 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,

More information

### Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes

Worksheet 6 th Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of

More information

### Here are two situations involving chance:

Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)

More information

### Stat210 WorkSheet#2 Chapter#2

1. When rolling a die 5 times, the number of elements of the sample space equals.(ans.=7,776) 2. If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet,

More information

### 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

More information

### 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

More information

### 1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.

1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find

More information

### 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

More information

### Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Diana Pell Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 1. An experiment

More information

### MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG

MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices

More information

### Probability. Chapter-13

Chapter-3 Probability The definition of probability was given b Pierre Simon Laplace in 795 J.Cardan, an Italian physician and mathematician wrote the first book on probability named the book of games

More information

### Chapter 16. Probability. For important terms and definitions refer NCERT text book. (6) NCERT text book page 386 question no.

Chapter 16 Probability For important terms and definitions refer NCERT text book. Type- I Concept : sample space (1)NCERT text book page 386 question no. 1 (*) (2) NCERT text book page 386 question no.

More information

### 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

More information

### 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

More information

### Probability. Engr. Jeffrey T. Dellosa.

Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional

More information

### , x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks)

1. The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). (1) Show that k = 3. Find E(X). (Total 7 marks) 2. In a game

More information

### Poker: Probabilities of the Various Hands

Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13

More information

### INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2

INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results

More information

### 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

More information

### Mathacle. Name: Date:

Quiz Probability 1.) A telemarketer knows from past experience that when she makes a call, the probability that someone will answer the phone is 0.20. What is probability that the next two phone calls

More information

### Empirical (or statistical) probability) is based on. The empirical probability of an event E is the frequency of event E.

Probability and Statistics Chapter 3 Notes Section 3-1 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful

More information

### 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

More information

### Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Diana Pell Section 11.1: The Fundamental Counting Principle Exercise 1. How many different two-letter words (including nonsense words) can be formed when

More information

### 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

More information

### Probability and Counting Techniques

Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each

More information

### 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

More information

### 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

More information

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

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

More information

### [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

More information

### Diamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES

CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times

More information

### Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include your name and student ID.

Math 3201 Unit 3 Probability Test 1 Unit Test Name: Part 1 Selected Response: Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include

More information

### Unit 9: Probability Assignments

Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose

More information

### INDIAN STATISTICAL INSTITUTE

INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 20-07-07 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it

More information

### Section The Multiplication Principle and Permutations

Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different

More information

### 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

More information

### 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

More information

### Poker: Probabilities of the Various Hands

Poker: Probabilities of the Various Hands 19 February 2014 Poker II 19 February 2014 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13

More information

### 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

More information

### Fundamentals of Probability

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

More information

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

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

More information

### CSC/MTH 231 Discrete Structures II Spring, Homework 5

CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the

More information

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is

More information

### 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

More information

### Name Class Date. Introducing Probability Distributions

Name Class Date Binomial Distributions Extension: Distributions Essential question: What is a probability distribution and how is it displayed? 8-6 CC.9 2.S.MD.5(+) ENGAGE Introducing Distributions Video

More information

### 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

More information

### 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

More information

### Chapter-wise questions. Probability. 1. Two coins are tossed simultaneously. Find the probability of getting exactly one tail.

Probability 1. Two coins are tossed simultaneously. Find the probability of getting exactly one tail. 2. 26 cards marked with English letters A to Z (one letter on each card) are shuffled well. If one

More information

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

More information