# Toss a coin twice. Let Y denote the number of heads.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 ! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i. The random variable is denoted by Y. The set of all possible values of Y is the set {y i = Y(e i ), i = 1, 2, 3 }. The set of all possible values of Y is a finite or countably infinite set, and Y is said to be a discrete random variable. Let y be the set of values of Y. The subset of all elementary events that are assigned the y, is the compound event {e i : Y(e i ) = y}. Usually the probabilities of all the elementary events are known, therefore the probability of this compound event can be readily computed by summing over all the elementary events. Toss a coin twice. Let Y denote the number of heads. Denote (Tail, Tail) to be the elementary event that the first toss is tail and the second toss is tail. Denote the other elementary events accordingly. Compound Event (Y=0) (Y=1) (Y=2) Elementary Events (Tail, Tail) (Tail, Head), (Head, Tail) (Head, Head)

2 Let Y be a discrete random variable. A probability mass function is f(y) = P{e i : Y(e i ) = y}. It is a function with values between 0 and 1 and whose sum is 1 over all values of y. Toss a balanced coin once. Let Y be the number of heads that occurs. Find the probability mass function of Y. Number of Heads Elementary Events (Y=0) (Tail) (Y=1) (Head) y f(y) 0 ½ 1 ½

3 Toss a balanced coin twice. Let Y denote the number of heads. Find the probability mass function of Y. Denote (Tail, Tail) to be the elementary event that the first toss is tail and the second toss is tail. Denote the other elementary events accordingly. Number of Heads (y) Elementary Events 0 (Tail, Tail) 1 (Tail, Head) (Head, Tail) 2 (Head, Head) " y f(y) 0 ¼ 1 ½ 2 ¼ A histogram is a graph of the probability mass function. The total area under a histogram is one. For a discrete random variable, the probability that Y is equal to y is the area in a histogram corresponding to a value y. f(y) y

4 Toss a pair of dice; win dollars equal to the sum of numbers on the two dice. Let Y denote the winnings after playing the game once. Find the probability mass function of Y. Winnings (y) Elementary Events 2 (1,1) 3 (1,2) (2,1) 4 (1,3) (2,2) (3,1) 5 (1,4) (2,3) (3,2) (4,1) 6 (1,5) (2,4) (3,3) (4,2) (5,1) 7 (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) 8 (2,6) (3,5) (4,4) (5,3) (6,2) 9 (3,6) (4,5) (5,4) (6,3) 10 (4,6) (5,5) (6,4) 11 (5,6) (6,5) 12 (6,6) y f(y) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/ / / /36

5 f(y) y

6 " #\$% & Take a production run of 100 machine parts, with 25 defective and 75 non-defective. Randomly draw two parts, replacing them after each draw. Let Y denote the number of defective parts drawn. Create a table for the random variable and the compound events. Find the probability mass function of Y. Draw the histogram. " #\$'% &#( Take a carton of 12 machine parts, with 3 defective and 9 non-defective. Randomly draw two parts for inspection, but do not replace them after each draw. Let Y denote the number of defective parts drawn. Create a table for the random variable and the compound events. Find the probability mass function of Y. Draw the histogram.

7 ! Take a carton of six machine parts, with one defective and five nondefective. Randomly draw parts for inspection, but do not replace them after each draw, until a defective part fails inspection. Let Y denote the number of parts drawn. Find the probability mass function of Y. Let F denote the event that a part fails inspection, and let P denote the event a part passes inspection. No. of Draws (y) Elementary Event Probability 1 (F) 1/6 2 (P, F) (5X1)/(6X5)=1/6 3 (P, P, F) (5X4X1)/(6X5X4)=1/6 4 (P, P, P, F) (5X4X3X1)/(6X5X4X3)=1/6 5 (P, P, P, P, F) (5X4X3X2X1)/(6X5X4X3X2)=1/6 6 (P, P, P, P, P, F) (5X4X3X2X1X1)/(6X5X4X3X2X1)=1/6 f(y) y

8 " Repeatedly toss a die until the number six comes up. Let Y denote the number of tosses. Find the probability mass function of Y. Let F denote the six appearing and let P the six not appearing. Imagine that a machine produces a defective part one time in six. Then Y is the number of the first defective part. In calculating the probabilities of the elementary events, we have used either equally likely events or the classical definition. (The classical definition uses ratio of the number ways an event can occur by the total number of ways any event can occur). Here we must use the special multiplicative rule of probability that determines the probability of the intersection of independent events as the product of the probabilities of the individual events. No. of Draws (y) Elementary Event Probability 1 (F) 1/6 = (P, F) (5/6)X(1/6) = (P, P, F) (5/6)X(5/6)X(1/6) = (P, P, P, F) (5/6)X(5/6)X(5/6)X(1/6) = (P, P, P, P, F) (5/6)X(5/6)X(5/6)X(5/6)X(1/6) = (P, P, P, P, P, F) (5/6)X(5/6)X(5/6)X(5/6)X(5/6)X(1/6) = f ( y) = 6 y = 1,2,... y 1 1 6

9 f(y) Note that this is an example of a countable infinite sample space. It is a special case of the geometric probability mass function. () * Let p be the probability that a manufactured part is defective. Let Y be the number of parts manufactured until the first defective part is produced. The probability mass function of Y is called the Geometric Distribution: f ( y) = p y = 1,2,... ( 1 p) y 1 (+ &*,+*- Let f(y) be the probability mass function of a random variable Y. The cumulative distribution function F(y) is F () y= PY ( y) = f() x x y The CDF F(y) is the area in the histogram up to y. The probability mass function may be recovered from the CDF: f () y= F() y F( y 1)

10 # The probability that the number of tosses is 3 or less is the cumulative distribution function evaluated at y = 3: F(3) = f(1) + f(2) + f(3) = = f(y) y y f(y) F(y)

11 F(y) The probability of exactly three tosses, f(3), may be obtained from the CDF: f(3) = F(3) F(2) = = " #\$ # Toss a pair of dice; win dollars equal to the sum of numbers on the two dice. Let Y denote the winnings after playing the game once. Construct the table of the values of the CDF. Graph the CDF.

### Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

### 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

### 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

### 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

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli

### Chapter 3 Random Variables and Probability Distributions

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

### 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,

### 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

### 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.

### 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

### Probability distributions

Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

### 2 Elementary probability

2 Elementary probability This first chapter devoted to probability theory contains the basic definitions and concepts in this field, without the formalism of measure theory. However, the range of problems

### Concepts of Probability

Concepts of Probability Trial question: we are given a die. How can we determine the probability that any given throw results in a six? Try doing many tosses: Plot cumulative proportion of sixes Also look

### 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

### Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

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

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

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

### Probability & Probability Distributions

Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions

### 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

### Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### 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

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### 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

### Solution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.3-4.4) Homework Solutions. Section 4.

Math 115 N. Psomas Chapter 4 (Sections 4.3-4.4) Homework s Section 4.3 4.53 Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give

### + Section 6.2 and 6.3

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

### Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

### 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

### P(X = x k ) = 1 = k=1

74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k

### Introduction to probability theory in the Discrete Mathematics course

Introduction to probability theory in the Discrete Mathematics course Jiří Matoušek (KAM MFF UK) Version: Oct/18/2013 Introduction This detailed syllabus contains definitions, statements of the main results

### E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

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

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

### 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

### 1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.

1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first

### 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

### Basic Probability Concepts

page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

### 10 GEOMETRIC DISTRIBUTION EXAMPLES:

10 GEOMETRIC DISTRIBUTION EXAMPLES: 1. Terminals on an on-line computer system are attached to a communication line to the central computer system. The probability that any terminal is ready to transmit

### The study of probability has increased in popularity over the years because of its wide range of practical applications.

6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,

### r r + b, r + k P r(r 1 ) = P r(r 2 R 1 ) = r + b + 2k r + 2k b r + b + 3k r(r + k)(r + 2k)b (r + b)(r + b + k)(r + b + 2k)(r + b + 3k).

Stat Solutions for Homework Set 3 Page 6 Exercise : A box contains r red balls and b blue balls. One ball is selected at random and its color is observed. The ball is then returned to the box and k additional

### 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

### 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

### Basic Probability Theory I

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

### Sample Questions for Mastery #5

Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### DISCRETE RANDOM VARIABLES

DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced

### 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.

### University of California, Los Angeles Department of Statistics. Random variables

University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.

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

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

### 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

### Models for Discrete Variables

Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

### Welcome to Stochastic Processes 1. Welcome to Aalborg University No. 1 of 31

Welcome to Stochastic Processes 1 Welcome to Aalborg University No. 1 of 31 Welcome to Aalborg University No. 2 of 31 Course Plan Part 1: Probability concepts, random variables and random processes Lecturer:

### 4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### Homework 6 (due November 4, 2009)

Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive

### I. WHAT IS PROBABILITY?

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

### Section 6-5 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)

### TEST 2 STUDY GUIDE. 1. Consider the data shown below.

2006 by The Arizona Board of Regents for The University of Arizona All rights reserved Business Mathematics I TEST 2 STUDY GUIDE 1 Consider the data shown below (a) Fill in the Frequency and Relative Frequency

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

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

### MAS108 Probability I

1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

### The random variable X - the no. of defective items when three electronic components are tested would be

RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS Example: Give the sample space giving a detailed description of each possible outcome when three electronic components are tested, where N - denotes nondefective

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### 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

### 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

### Worked examples Basic Concepts of Probability Theory

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

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

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

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### Statistics 100A Homework 3 Solutions

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

### 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

### 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

### Introduction to Probability

Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

### 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

### IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics

### 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

### Jan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 50-54)

Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0- Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample

### Chapter 4: Probability and Counting Rules

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

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### Ch. 13.2: Mathematical Expectation

Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we

### Pattern Recognition. Probability Theory

Pattern Recognition Probability Theory Probability Space 2 is a three-tuple with: the set of elementary events algebra probability measure -algebra over is a system of subsets, i.e. ( is the power set)

### 6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS Certain derivations presented in this course have been somewhat heavy on algebra. For example, determining the expectation of the Binomial distribution (page 5.1 turned

### 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

### MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

### CONTINGENCY (CROSS- TABULATION) TABLES

CONTINGENCY (CROSS- TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods

### 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

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

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

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

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

### The 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

### Study Manual for Exam P/Exam 1. Probability

Study Manual for Exam P/Exam 1 Probability Eleventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com

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

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

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

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

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

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

### Probability 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

### For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )

Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll

### Study Manual for Exam P/Exam 1. Probability

Study Manual for Exam P/Exam 1 Probability Seventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com

### 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

### 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