A Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability - 1. Probability and Counting Rules

Size: px
Start display at page:

Download "A Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability - 1. Probability and Counting Rules"

Transcription

1 Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample have the disease, what does it mean? How likely would this happen if the researcher is right? Simple Example What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). So toss a coin twice. Do it! Did you get one head & one tail? What s it all mean? 1 2 Sample Space and Probability Random Experiment: (Probability Experiment) an experiment whose outcomes depend on chance. Sample Space (S): collection of all possible outcomes in random experiment. Event (E): a collection of outcomes 3 Sample Space and Event Sample Space: S {Head, Tail} S {Life span of a human} {x x 0, x R} S {ll individuals live in a community with or without having disease } Event: E {Head} E {Life span of a human is less than 3 years} E {The individuals who live in a community that has disease } 4 Sample Space {(H,1),(H,2),(H,3),(H,4),(T,1),(T,2),(T,3),(T,4)} Tree Diagram H T 2 x 4 1 (H,1) 2 (H,2) 3 (H,3) 4 (H,4) 1 (T,1) 2 (T,2) 3 (T,3) 4 (T,4) Tree Diagram What is the sample space for casting two dice experiment? Die 2 1 (1,1) Die x 6 36 outcomes (1,2) (1,3) (1,4) (1,) (1,6) 6 Probability - 1

2 Sample Space (Two Dice) Compound event Sample Space: S {(1,1),(1,2),(1,3),(1,4),(1,),(1,6), (2,1),(2,2),(2,3),(2,4),(2,),(2,6), (3,1),(3,2),(3,3),(3,4),(3,),(3,6), (4,1),(4,2),(4,3),(4,4),(4,),(4,6), (,1),(,2),(,3),(,4),(,),(,6), (6,1),(6,2),(6,3),(6,4),(6,),(6,6)} Simple event How many outcomes are Sum is 7? 7 Definition of Probability rough definition: (frequentist definition) Probability of a certain outcome to occur in a random experiment is the proportion of times that the this outcome would occur in a very long series of repetitions of the random experiment. Total Heads / Number of Tosses Number of Tosses Determining Probability How to determine probability? Empirical Probability Theoretical Probability (Subjective approach) Empirical Probability ssignment Empirical study: (Don t know if it is a balanced Coin?) Outcome Head Tail Total Empirical Probability ssignment Empirical probability assignment: Empirical Probability Distribution Empirical study: Number of times event E occurs E) Number of times experiment is repeated m n Probability of Head: Head) 12 Outcome Head % Empirical Probability Distribution Tail Total Probability Probability - 2

3 Theoretical Probability ssignment Make a reasonable assumption: What is the probability distribution in tossing a coin? ssumption: We have a balanced coi Theoretical Probability ssignment Theoretical probability assignment: Number of equally likely outcomes in event E E) Size of the sample space E) S) Probability of Head: 1 Head) 2. 0% Theoretical Probability Distribution Empirical study: Outcome Head Tail Total Probability Probability Distribution (Probability Model) If a balanced coin is tossed, Head and Tail are equally likely to occur, Head). 1/2 and Tail). 1/2 1/2 Probability ar Chart Total probability is 1. Empirical Probability Distribution 1 Head Tail 16 Relative and Probability Distributions Discrete Distribution Number of times visited a doctor from a random sample of 300 individuals from a community Class Relative ) ) ) ) ) ).01 Total Relative Distribution Probability - 3

4 Relative and Probability When selecting one individual at random from a population, the probability distribution and the relative frequency distribution are the same. 19 Probability for the Discrete Case If an individual is randomly selected from this group 300, what is the probability that this person visited doctor 3 times? 3 times) (42)/300 Class Relative Total or 14% 20 Discrete Distribution If an individual is randomly selected from this group 300, what is the probability that this person visited doctor 4 or times? Class Total or times) 4) + ) Relative It would be an empirical probability distribution, if the sample of 300 individuals is utilized for understanding a large population. 21 Properties of Probability Probability is always a value between 0 and 1. Total probability (all outcomes together) equals 1. Probability of either one of the disjoint events or to occur is the sum of their individual probabilities. or ) ) + ) 22 Probability Distribution (Probability Model) Complementation Rule If a balanced coin is tossed, Head (H) and Tail (T) are equally likely to occur, For any event E, all possible outcomes) H) + T) Probability Total probability is 1 ar Chart 1/2 H). and T). Head Tail 23 E does not occur) 1 E) Complement of E E * Some places use E c or E EE) E) E 24 Probability - 4

5 Complementation Rule Complementation Rule If an unbalanced coin has a probability of 0.7 to turn up Head each time tossing this coin. What is the probability of not getting a Head for a random toss? not getting Head) If the chance of a randomly selected individual living in community to have disease H is.001, what is the probability that this person does not have disease H? having disease H).001 not having disease H) 1 having disease H) Topics Rules of Probability Notations and Venn Diagram ddition Rules Conditional Probability Multiplication Rules Odds Ratio & Relative Risk 27 S {1, 2, 3, 4,, 6} {1, 2, 3} {3, 6} Intersection of events: <> and Example: {3} Union of events: <> or Example: {1, 2, 3, 6} S Venn Diagram (with elements listed) 28 ddition Rules for Probability Mutually Exclusive Events and are mutually exclusive (disjointed) events ddition Rule 1 (Special ddition Rule) In an experiment of casting an unbalanced die, the event and are defined as the following: an odds number 6 and given ).61, ) 0.1. What is the probability of getting an odds number or a six? and are mutually exclusive dditive Rule: ) ) + ) 29 ) ) + ) Probability -

6 ddition Rule 1 - generalized (for multiple events) In 1, 2,,, k are defined k mutually exclusive (m.e.) then 1 2 k ) 1 )+ 2 )+ + k ) Example: When casting a balanced die, the probability of getting 1 or 2 or 3 is 1 2 3) 1) + 2) + 3) 1/6 + 1/6 + 1/6 1/2 31 Example: When casting a unbalanced die, such that 1).2, 2).1, 3).3, the probability of getting 1 or 2 or 3 is 1 2 3) 1) + 2) + 3) General ddition Rule Not M.E.! Venn Diagram (with counts) Given total of 100 subjects )? Sample Space? 4 ) ) + ) ) 33 Smokers, ) 0 Lung Cancer, ) 2 ) 20 Joint Event 34 Venn Diagram (with relative frequencies) Given a sample space ).?.4 Probability of Joint Events In a study, an individual is randomly selected from a population, and the event and are defined as the followings: the individual is a smoker. the individual has lung cancer ).0, ).2 and ) ) smoker and has lung cancer) Smokers, ).0 Lung Cancer, ).2 ).20 Joint Event 3 What is the probability or ) )? 36 Probability - 6

7 ddition Rule 2 (General ddition Rule) What is the probability that this randomly selected person is a smoker or has lung cancer? or )? or ) ) + ) - and ) Contingency Table Cancer, No Cancer, c Total Smoke, Not Smoke, c Venn Diagram Conditional Probability The conditional probability of event to occur given event has occurred (or given the condition ) is denoted as ) and is, if ) is not zero, E) # of equally likely outcomes in E, ) ) ) or ) ) ) Conditional Probability Smoke S Not Smoke S Cancer C 20 2 No Cancer C ) ) ) Total C S) 20/0.4 C S ' ) / Conditional Probability Smoke S Not Smoke S Cancer C 20 (.2) (.0) 2 C) (.2) C S).2/..4 C S ' ).0/..1 No Cancer C 30 (.3) 4 (.4) 7 C)(.7) ) ) ) Total 0 S) (.) 0 S ) (.) 100 (1.0) C S) What is C S )? 4 (Relative Risk ) 41 Independent Events Events and are independent if ) ) or ) ) or and ) ) ) 42 Probability - 7

8 Multiplication Rule 1 (Special Multiplication Rule) If event and are independent, and ) ) ) Example If a balanced die is rolled twice, what is the probability of having two 6 s? 6 1 the event of getting a 6 on the 1st trial 6 2 the event of getting a 6 on the 2nd trial 6 1 ) 1/6, 6 2 ) 1/6, 6 1 and 6 2 are independent events 6 1 and 6 2 ) 6 1 ) 6 2 ) (1/6)(1/6) 1/ Independent Events 10% of the people in a large population has disease H. If a random sample of two subjects was selected from this population, what is the probability that both subjects have disease H? H i : Event that the i-th randomly selected subject has disease H. H 2 H 1 ) H 2 ) [Events are almost independent] H 1 H 2 )? H 1 ) H 2 ).1 x Independent Events If events 1, 2,, k are independent, then 1 and 2 and and k ) 1 ) 2 ) k ) What is the probability of getting all heads in tossing a balanced coin four times experiment? H 1 ) H 2 ) H 3 ) H 4 ) (.) Independent Events 10% of the people in a large population has disease H. If a random sample of 3 subjects was selected from this population, what is the probability that all subjects have disease H? H i : Event that the i-th randomly selected subject has disease H. [Events are almost independent] H 1 H 2 H 3 )?H 1 ) H 2 ) H 3 ).1 x.1 x Multiplication Rule 2 (General Multiplication Rule) For any two events and, and ) ) ) ) ) ) ) and ) ) and ) ) 48 Probability - 8

9 Multiplication Rule 2 Multiplication Rule 2 If in the population, 0% of the people smoked, and 40% of the smokers have lung cancer, what percentage of the population that are smoker and have lung cancer? S) 0% of the subjects smoked C S) 40% of the smokers have cancer C and S) C S) S).4 x ox 1 contains 2 red balls and 1 blue ball ox 2 contains 1 red ball and 3 blue balls coin is tossed. If Head turns up a ball is drawn from ox 1, and if Tail turns up then a ball is drawn from ox 2. Find the probability of selecting a red ball. H and Red) + T and Red) H)Red H) + T)Red T) (1/2) x (2/3) + (1/2) x (1/4) 1/3 + 1/8 11/24 0 Counting Rules Fundamental Principle of Counting Counting Rule: (use of the notation k n ) In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so forth, the total possibilities of the sequence will be k 1 k 2 k 3 k n 2 x 4 8 possible outcomes (coin & wheel) 6 X 6 36 possible outcomes (2 dice) k 1 k 2 Example: There are three shirts of different colors, two jackets of different styles and five pairs of pants in the closet. How many ways can you dress yourself with one shirt, one jacket and one pair of pants selected from the closet? Sol: 3 x 2 x 30 ways 1 K 1 x k 2 x k 3 2 Permutations How many different four-letter code words can be formed by using the four letters in the word SE without repeating use of the same letter? k 1 4 k 2 3 k 3 2 k S E Factorial Formula For any counting number n 1 x 2 x x (n 1) x n 0! 1 Example: 3! 1 x 2 x Probability - 9

10 How many different four-letter code words can be formed by using the four letters in the word SE and the same letter can be used repeatedly? k 1 4 k 2 4 k 3 4 k S E How many different four-letter code words can be formed by using the four letters selected from letters through Z and the same letter can not be used repeatedly? k 1 26 k 2 2 k 3 24 k ,800 S E 6 How many different four-letter code words can be formed by using the four letters selected from letters through Z and the same letter can be used repeatedly? k 1 26 k 2 26 k 3 26 k ,976 S E Permutation Rule Permutation Rule: The number of possible permutations of r objects from a collection of n distinct objects is P r n ( n r)! Order does count! 7 8 Permutations P r n ( n r)! Combination Rule How many ways can a four-digit code be formed by selecting 4 distinct digits from nine digits, 1 through 9, without repeating use of the same digit? 9 P 4 9! 9! ! (9-4)!!! Combination Rule: The number of possible combinations of r objects from a collection of n distinct objects is C r n Order does not count! r!( n r)! 60 Probability - 10

11 Combinations C r n r!( n r)! How many ways can a committee be formed by selecting 3 people from a group 10 candidates? n 10, r 3 10! 10! ! 10C 3 3! (10-3)! 3! 7! 3! 7! ! Combinations How many ways can a combination of 4 distinct digits be selected from nine digits, 1 through 9? 9 C 4 9! 4! (9-4)! ! 4!! C r n r!( n r)! ! 4!! C, C, C, C, C, C are the same combination Combinations How many ways can 6 distinct numbers be selected from a set of 47 distinct numbers? n 47, r 6 47! ! 47C 6 6! 41! 6! 41! ! 10,737,73 C r n r!( n r)! 63 Probability Using Counting (Theoretical pproach) In a lottery game, one selects 6 distinct numbers from a set of 47 distinct numbers, what is the probability of winning the jackpot? n 47, r 6 47C 6 47!/(6! 41!) 10,737,73 Jackpot) 1 10,737,73 (Theoretical Prob.) 64 irthday Problem In a group of randomly select 23 people, what is the probability that at least two people have the same birth date? (ssume there are 36 days in a year.) at least two people have the same birth date) Too hard!!! 1 everybody has different birth date) 1 [36x364x x( )] / inomial Probability What is the probability of getting two 6 s in casting a balanced die times experiment? S S S S S ) (1/6) 2 x (/6) S S S S S ) (1/6) 2 x (/6) 3 S S S S S ) (1/6) 2 x (/6) 3! How many of them? !3! Probability (two 6 s) x Probability - 11

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

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

+ Section 6.2 and 6.3

+ 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

More information

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. 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

More information

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

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

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

Basic Probability Theory I

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

More information

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

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

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

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 4: Probability and Counting Rules

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

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

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

Basic concepts in probability. Sue Gordon

Basic concepts in probability. Sue Gordon Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

More information

Probability OPRE 6301

Probability OPRE 6301 Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

More information

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

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

More information

STAT 35A HW2 Solutions

STAT 35A HW2 Solutions STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

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

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

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

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

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

Math 150 Sample Exam #2

Math 150 Sample Exam #2 Problem 1. (16 points) TRUE or FALSE. a. 3 die are rolled, there are 1 possible outcomes. b. If two events are complementary, then they are mutually exclusive events. c. If A and B are two independent

More information

Chapter 2: Systems of Linear Equations and Matrices:

Chapter 2: Systems of Linear Equations and Matrices: At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,

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

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

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,

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

More information

Worked examples Basic Concepts of Probability Theory

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

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

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

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

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

More information

Chapter 3: The basic concepts of probability

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

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

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

Ch. 12.1: Permutations

Ch. 12.1: Permutations Ch. 12.1: Permutations The Mathematics of Counting The field of mathematics concerned with counting is properly known as combinatorics. Whenever we ask a question such as how many different ways can we

More information

Counting principle, permutations, combinations, probabilities

Counting principle, permutations, combinations, probabilities Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing

More information

Fundamentals of Probability

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

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

Probability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)

Probability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability

More information

Session 8 Probability

Session 8 Probability Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

More information

Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A.

Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. Complement If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. For example, if A is the event UNC wins at least 5 football games, then A c is the

More information

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

More information

Probability Using Dice

Probability Using Dice Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you

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

I. WHAT IS PROBABILITY?

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

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

Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

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

4.4 Conditional Probability

4.4 Conditional Probability 4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

More information

MAT 1000. Mathematics in Today's World

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

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

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. Ch. - Problems to look at Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability

More information

Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13. Understanding Probability and Long-Term Expectations Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

More information

Elements of probability theory

Elements of probability theory The role of probability theory in statistics We collect data so as to provide evidentiary support for answers we give to our many questions about the world (and in our particular case, about the business

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant

More information

Topic 1 Probability spaces

Topic 1 Probability spaces CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in five-card poker is about 0.2%,

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

E3: PROBABILITY AND STATISTICS lecture notes

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............................

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

Unit 4 The Bernoulli and Binomial Distributions

Unit 4 The Bernoulli and Binomial Distributions PubHlth 540 4. Bernoulli and Binomial Page 1 of 19 Unit 4 The Bernoulli and Binomial Distributions Topic 1. Review What is a Discrete Probability Distribution... 2. Statistical Expectation.. 3. The Population

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

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

The Casino Lab STATION 1: CRAPS

The Casino Lab STATION 1: CRAPS The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will

More information

Introduction to Probability

Introduction to Probability 3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which

More information

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

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

6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers 6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

More information

11-1 Permutations and Combinations

11-1 Permutations and Combinations Fundamental Counting Principal 11-1 Permutations and Combinations Using the Fundamental Counting Principle 1a. A make-your-own-adventure story lets you choose 6 starting points, gives 4 plot choices, and

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

Question 1 Formatted: Formatted: Formatted: Formatted:

Question 1 Formatted: Formatted: Formatted: Formatted: In many situations in life, we are presented with opportunities to evaluate probabilities of events occurring and make judgments and decisions from this information. In this paper, we will explore four

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

MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

More information

2 Elementary probability

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

More information

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values. MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum

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

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

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

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

Probability distributions

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,

More information

Responsible Gambling Education Unit: Mathematics A & B

Responsible Gambling Education Unit: Mathematics A & B The Queensland Responsible Gambling Strategy Responsible Gambling Education Unit: Mathematics A & B Outline of the Unit This document is a guide for teachers to the Responsible Gambling Education Unit:

More information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

More information

TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities

TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE Roulette is one of the most popular casino games. The name roulette is derived from the French word meaning small

More information

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

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

Basic Probability Concepts

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

More information

A probability experiment is a chance process that leads to well-defined outcomes. 3) What is the difference between an outcome and an event?

A probability experiment is a chance process that leads to well-defined outcomes. 3) What is the difference between an outcome and an event? Ch 4.2 pg.191~(1-10 all), 12 (a, c, e, g), 13, 14, (a, b, c, d, e, h, i, j), 17, 21, 25, 31, 32. 1) What is a probability experiment? A probability experiment is a chance process that leads to well-defined

More information

IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION

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

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

Algebra 2 C Chapter 12 Probability and Statistics

Algebra 2 C Chapter 12 Probability and Statistics Algebra 2 C Chapter 12 Probability and Statistics Section 3 Probability fraction Probability is the ratio that measures the chances of the event occurring For example a coin toss only has 2 equally likely

More information

Study Manual for Exam P/Exam 1. Probability

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

More information

Combinations and Permutations

Combinations and Permutations Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination

More information

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

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

More information

https://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10

https://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10 1 of 8 4/9/2013 8:17 AM PRINTABLE VERSION Quiz 10 Question 1 Let A and B be events in a sample space S such that P(A) = 0.34, P(B) = 0.39 and P(A B) = 0.19. Find P(A B). a) 0.4872 b) 0.5588 c) 0.0256 d)

More information

MAS113 Introduction to Probability and Statistics

MAS113 Introduction to Probability and Statistics MAS113 Introduction to Probability and Statistics 1 Introduction 1.1 Studying probability theory There are (at least) two ways to think about the study of probability theory: 1. Probability theory is a

More information

Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS

Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS 1. Give two examples of ways that we speak about probability in our every day lives. NY REASONABLE ANSWER OK. EXAMPLES: 1) WHAT

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

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

Chapter 3. Probability

Chapter 3. Probability Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.

More information

What is the purpose of this document? What is in the document? How do I send Feedback?

What is the purpose of this document? What is in the document? How do I send Feedback? This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Statistics

More information