Probability definitions


 Sandra Polly Russell
 1 years ago
 Views:
Transcription
1 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 datagenerating process Examples: toss a coin (one or more times), roll 2 dice, select 5 cards from a deck, interview 100 people for market research, observe the reaction of 50 patients to a new drug. 3. Sample space = set of all possible outcomes of an experiment. Example: if two dice (a red die and a green die) are tossed, any outcome is described by the number on the red die and the number on the green die. Note that the outcome is a low level description of what happened. Let the number on the red die be indicated by bold italics, so that (3, 2) indicates a 3 on the red die and a 2 on the green die. In the experiment of tossing a red die and a green die, we can list the sample space in a table: 1, 1 1, 2 1, 3 1, 4 1, 5 1, 6 2, 1 2, 2 2, 3 2, 4 2, 5 2, 6 3, 1 3, 2 3, 3 3, 4 3, 5 3, 6 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 5, 1 5, 2 5, 3 5, 4 5, 5 5, 6 6, 1 6, 2 6, 3 6, 4 6, 5 6, 6 4. Event = any collection or subset of outcomes in the sample space. Events are simple if they contain exactly one outcome, and are compound if they contain more than one. Example: We can define the event A = both dice show the same number, and calculate P(A) = 6 / 36. (If you xerox or copy the sheet, shade in the appropriate areas) Define event B = sum of two numbers at least 10. P (B) = 6 / Random variables associate a number with an event. If we define the random variable X as the sum of the two numbers on the dice, we can say P (X = 6) = 5/36 6. Union of two events, denoted by as in A B, constructs a new event both dice show the same number, or the sum of the two numbers is at least 10. This should be read as A or B. Find the probability of A or B in the example above. Does P(A or B) = P(A) + P(B)? Why not? 7. Intersection of two events, denoted by as in A B, constructs a new event  both dice show the same number, which is greater than or equal to 10. This should be read as A and B. Find the probability of A and B in the example above. Does P (A and B) = P(A) times P(B)? Why not? 8. Complement of an event A, denoted by a prime or superscript c, as in A or A c, indicates those outcomes not in the event A. What is the probability of A in the example above? Note that P(A ) = P(A) This relationship often makes calculations much simpler, especially when the problem includes phrases such as at least or at most. Example (Chevalier de la Mere): what is the probability that at least one six turns up in 4 tosses of a die? [Hint: it is a little more than half, De la Mere found that out by extensive experiment.] What is the probability that at least one double six turns up in 24 tosses of two dice? [The chance of double six is 1/36, but we compensate by having 6 times as many tosses, so de la Mere thought the probability should be the same. Is it? Hint: de la Mere lost a lot of money by believing this]
2 Statistics 1040 Dr. McGahagan Probability problems Simple occurences: Event Get a tail in a single toss of a fair coin Roll a 3 on a normal, 6 sided die Probability Draw a heart from a deck or cards Child born on a Saturday or Sunday More complicated events, for which it will be helpful to use random variables to keep track of outcomes (for example, one might want the random variable X = number of heads in 3 tosses of a fair coin) Event Probability Toss 3 heads in 3 tosses of a coin Roll boxcars (6 on each of 2 dice) Be dealt a flush (5 cards of same suit) in a standard 5 card deal. Toss exactly 2 heads and two tails IN THAT ORDER in 4 tosses of a coin Toss exactly 2 heads and two tails IN ANY ORDER in 4 tosses of a coin Roll either boxcars or snakeeyes (2 sixes or 2 ones in a roll of 2 dice.
3 Addition and multiplication rules the full story In some of the above problems, we used the addition rule in the form P(A) or P(B) = P(A) + P(B) and the multiplication rule in the form P(A and B) = P(A) P(B) We must extend the rules to take account of 1. events that are not mutually exclusive that is, events which can both happen at the same time. Suppose you have been dealt a poker hand of Jack,Queen, King and Ace of hearts along with the 2 of clubs. You are interested in the change of coming up with a winning hand if you discard the 2 and draw another card. What are the chances of getting either a 10 or a heart? We can assume that either a flush or a straight will win but there is the chance here of getting a royal flush with the 10 of hearts. 2. Events that are not independent that is, in which the probability of one event affects the probability of another. What is the chance of drawing two hearts in a row? Hint: NOT 13/52 times 13/52. Why? Suppose the chance that, for the US as a whole, the chance that a family s first car is domestic is.75, denoted as P(D1) =.75, and the chance that the second car is domestic is.4, denoted as P(D2) =.4 What is the chance that both cars are domestic? If the two events are independent, we could apply the multiplication rule P(D1 and D2) = P(D1) x P(D2) = 0.75 x 0.4 = 0.30 But it may be that purchasers show buyer loyalty, that is, those who purchased a domestic car for their first car are more likely than the average to buy a domestic car for their second car. Assume that P(D2 given D1) = 0.6 also written P(D1 D2) = 0.6 and calculate P(D1 and D2). Hint: look back at the two hearts in a row problem. P (D1 and D2) = P(D1) x P (D2 D1) =.75 x 0.6 = 0.45
4 A visual presentation of a similar problem: D1 = a family's first car is domestic; F1 = probability first car is foreign D2 = a family's second car is domestic; F2 = probability second car is foreign Given that P(D1) = 0.75 and P(D2) = 0.4, and assuming there are 100 total cars and that P(D1 and D2) = 0.35 fill in the following table: A few numbers have been filled in to get you started. Be sure you understand how they were arrived at, and how they reflect the statements above. Car 2 is domestic Car 2 is foreign ROW TOTALS Car 1 is domestic Car 1 is foreign COLUMN totals After doing so, calculate all the joint probabilities: P(D1 and D2) = 0.35 P(D1 and F2) = P(F1 and D2) = P(F1 and F2) = And all the conditional probabilities P(D2 D1) = P(D2 F1) = P(F2 D1) = P(F2 F1) = What is the probability that (for two car families described in the table above): a. Both a family's cars are domestic? b. Both a family's cars are foreign? c. A family has one domestic and one foreign car? d. We know that the Smith family has at least one domestic car. What is the chance that they also have a foreign car? e. We know that the Jones family has at least one foreign car. What is the chance that they also have a domestic car?
5 Answers: Car 2 is domestic Car 2 is foreign ROW TOTALS Car 1 is domestic Car 1 is foreign COLUMN totals The JOINT PROBABILITIES are easy: The table gives the NUMBERS of families in each category  there are 35 families with both car 1 and car 2 being domestic, so the probability that any twocar family chosen at random having two domestic cars is 35 / 100 = 0.35 or 35 percent. P (D1 and D2 ) = 0.35 (answer for [a] on last page) P (D1 and F2) = 0.40 P (F1 and D2) = 0.05 P (F1 and F2) = 0.20 (answer for [b] on last page) For [c] on the previous page, note that families will have one domestic and one foreign car if their two cars are in the square D1 and F2 OR in the square F1 and D2. The OR is telling us to ADD the joint probabilities P(D1 and F2) + P (F1 and D2) = = 0.45 The ROW and COLUMN totals give the MARGINAL PROBABILITIES, since they are written in the margin of the detailed table (don't think of marginal cost!). P(D1) = 0.75 P (F1) = 0.40 P (D2) = 0.25 P (F2) = 0.60 CONDITIONAL PROBABILITIES can be read off the table: If you are GIVEN that D1 is domestic, you know you are in the first ROW of the table  the information given means the family is one of the 75 for whom the first car is domestic. You can mentally reduce the entire table to: Car 2 is domestic Car 2 is foreign ROW TOTALS Car 1 is domestic Hence the probability that their second car is foreign is: P (F2 D1) = 40 / 75 = If we are GIVEN that the second car is domestic, the probability that the first car is foreign is P (F1 D2) = 5 / 40 = (mentally reduce the entire table to the first column) If we are given (as in part [d]) that the Smith family has at least one domestic car, we delete the F1  F2 square from the table, leaving 80 families, so chance of also having a foreign car is 45 / 80 = For the Jones family in part [e], with at least one foreign car, delete the D1D2 square; their chance of also having a domestic car is 20 / 65 =
6 Bayes's Theorem Suppose that 80 percent of the taxicabs in town are owned by the Yellow Cab Company and 20 percent are owned by the Blue Cab Company, and that they are painted accordingly. A taxicab driver was arrested in a bank robbery. A witness claims that a cab was used as the getaway car, and thinks that the cab was blue, although he admits that the light was poor. As a result of repeated tests in similar lighting conditions, the defense finds that the witness is 75 percent accurate  that he correctly identifies a blue cab as blue 75 percent of the time, but incorrectly identifies a blue cab as yellow 25 percent of the time. We should: (a) reject the witness testimony as not perfectly accurate, and treat the probability of the cab being blue as 20 percent. (b) accept the witness as having a 75 percent chance of being right and the cab being blue (c) treat the probability of the cab being blue as somewhere between 20 and 75, but closer to 20 (that is, more than 20 but less than 47.5) (d) treat the probability of the cab being blue as somewhere between 20 and 75, but closer to 75 (that is, more than 47.5 but less than 75) Answer: The answer will be between the two extremes  although not perfectly accurate, the witness is right more often than not, and his claim that the cab is blue raises the chance to more than 20 percent. But it is NOT 75 percent: the test establishes the chance the witness SAYS the cab is blue, GIVEN THAT it is in fact blue at 75 percent, but we are interested in the "inverse probability" that the cab is REALLY blue, GIVEN that the witness SAYS it is blue. P (Says B B) = 0.75 P (Says B B) = P (B and SAYS B) / P (B) from the definition of conditional probability P (B Says B) = P (B and SAYS B) / P (Says B) has the same numerator, but a different denominator. It is a simple application of the multiplication rule for dependent events to compute the numerator: P (B and Says B) = P (B) * P (Says B B) = 0.20 * (.75) = 0.15 Make a table and fill in the other possibilites: P (B and Says Y) = P (B) * P (Says Y B) = 0.20 * (.25) = 0.05 P (Y and Says B) = P (Y) * P (Says B Y) = 0.80 * (.25) = 0.20 P (Y and Says Y) = P (Y) * P (Says Y Y) = 0.80 * (.75) = 0.60 Says B Says Y Row totals IS B IS Y Col. totals [Grand total = 1.00 or 100 percent] We know the witness said B, so we know that the first column is the only one that counts. Note that despite his 75 percent accuracy, the fact that there are more yellow cabs means that our witness makes more mistakes than correct identifications. P (B Says B) = 15 / 35 = 3/7 = = percent
+ 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 informationA (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.
Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome
More information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More informationRemember to leave your answers as unreduced fractions.
Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,
More informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 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
More informationMATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics
MATH 10: Elementary Statistics and Probability Chapter 3: Probability Topics Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you should
More informationEvents. Independence. Coin Tossing. Random Phenomena
Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,
More informationAn event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event
An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the
More informationWhat is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts
Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called
More informationQuestion: What is the probability that a fivecard 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 informationPROBABILITY. Chapter Overview
Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability
More informationLesson 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 information4.4 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 informationStatistics 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 informationDefinition 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 informationMassachusetts Institute of Technology
n (i) m m (ii) n m ( (iii) n n n n (iv) m m Massachusetts Institute of Technology 6.0/6.: Probabilistic Systems Analysis (Quiz Solutions Spring 009) Question Multiple Choice Questions: CLEARLY circle the
More informationConsider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.
Probability and the ChiSquare Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system
More informationPROBABILITY NOTIONS. Summary. 1. Random experiment
PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non
More informationAP 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 informationSection 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 informationProbability. 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 informationMAT 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 informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationStatistical 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 informationBasic 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 informationFor 2 coins, it is 2 possible outcomes for the first coin AND 2 possible outcomes for the second coin
Problem Set 1. 1. If you have 10 coins, how many possible combinations of heads and tails are there for all 10 coins? Hint: how many combinations for one coin; two coins; three coins? Here there are 2
More informationChapter 5  Probability
Chapter 5  Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set
More informationProbability Review. ICPSR Applied Bayesian Modeling
Probability Review ICPSR Applied Bayesian Modeling Random Variables Flip a coin. Will it be heads or tails? The outcome of a single event is random, or unpredictable What if we flip a coin 10 times? How
More informationSection 65 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 informationBasic 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 informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationThe 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 informationMATH 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 informationExample: If we roll a dice and flip a coin, how many outcomes are possible?
12.5 Tree Diagrams Sample space Sample point Counting principle Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible
More informationBasic 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 informationMath 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 informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationChingHan Hsu, BMES, National Tsing Hua University c 2014 by ChingHan Hsu, Ph.D., BMIR Lab
Lecture 2 Probability BMIR Lecture Series in Probability and Statistics ChingHan Hsu, BMES, National Tsing Hua University c 2014 by ChingHan Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random
More informationDiscrete 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 information4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style
Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring
More informationMATH 201. Final ANSWERS August 12, 2016
MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6sided dice, five 8sided dice, and six 0sided dice. A die is drawn from the bag and then rolled.
More informationProbability. Vocabulary
MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability
More informationSTAT 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 information1 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 informationLecture 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 informationCombinations 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 informationBasic 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 informationBasics of Probability
Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 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 information7.1 Sample space, events, probability
7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationJan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 5054)
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 information94 Counting Solutions for Chapter 3. Section 3.2
94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length4 lists are there? Answer: 6 6 6 6 = 1296. (b
More informationMath/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 informationCh. 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 informationChapter 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 informationProbability 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 informationLecture 2: Probability
Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability
More informationALevel Maths. in a week. Core Maths  Coordinate Geometry of Circles. Generating and manipulating graph equations of circles.
ALevel Maths in a week Core Maths  Coordinate Geometry of Circles Generating and manipulating graph equations of circles. Statistics  Binomial Distribution Developing a key tool for calculating probability
More information6.3 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 information2.5 Conditional Probabilities and 2Way Tables
2.5 Conditional Probabilities and 2Way Tables Learning Objectives Understand how to calculate conditional probabilities Understand how to calculate probabilities using a contingency or 2way table It
More informationDiscrete Mathematics Lecture 5. Harper Langston New York University
Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = 1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of
More informationDiscrete 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 informationConditional Probability and General Multiplication Rule
Conditional Probability and General Multiplication Rule Objectives:  Identify Independent and dependent events  Find Probability of independent events  Find Probability of dependent events  Find Conditional
More informationUnit 19: Probability Models
Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,
More informationDiscrete 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 informationDistributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment
C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning
More informationIntroduction and Overview
Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering
More informationAll You Ever Wanted to Know About Probability Theory, but Were Afraid to Ask
All You Ever Wanted to Know About Probability Theory, but Were Afraid to Ask 1 Theoretical Exercises 1. Let p be a uniform probability on a sample space S. If S has n elements, what is the probability
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationProbability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.
Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111
More informationThe 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 informationReview of Probability
Review of Probability Table of Contents Part I: Basic Equations and Notions Sample space Event Mutually exclusive Probability Conditional probability Independence Addition rule Multiplicative rule Using
More informationLesson 3 Chapter 2: Introduction to Probability
Lesson 3 Chapter 2: Introduction to Probability Department of Statistics The Pennsylvania State University 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of
More informationAlg2 Notes 7.4.notebook February 15, Two Way Tables
7 4 Two Way Tables Skills we've learned 1. Find the probability of rolling a number greater than 2 and then rolling a multiple of 3 when a number cube is rolled twice. 2. A drawer contains 8 blue socks,
More information1. 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 informationCurriculum Design for Mathematic Lesson Probability
Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.
More informationIntroduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang
Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 1 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space
More information4.5 Finding Probability Using Tree Diagrams and Outcome Tables
4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or
More informationPROBABILITY. 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 informationToss a coin twice. Let Y denote the number of heads.
! 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.
More informationSection Tree Diagrams. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 12.5 Tree Diagrams What You Will Learn Counting Principle Tree Diagrams 12.52 Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed
More informationAn approach to Calculus of Probabilities through real situations
MaMaEuSch Management Mathematics for European Schools http://www.mathematik.unikl.de/ mamaeusch An approach to Calculus of Probabilities through real situations Paula Lagares Barreiro Federico Perea RojasMarcos
More informationIt is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important
PROBABILLITY 271 PROBABILITY CHAPTER 15 It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important subject of human knowledge.
More informationChapter 5: Probability: What are the Chances? Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation
Chapter 5: Probability: What are the Chances? Section 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 5 Probability: What Are
More informationProbability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.
Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how
More information34 Probability and Counting Techniques
34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.
More informationFind 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 informationI. 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 informationLab 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
More informationMath 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141
Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall
More informationProbability: 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 informationCHAPTER 5 Probability: What Are the Chances?
CHAPTER 5 Probability: What Are the Chances? 5.1 Randomness, Probability, and Simulation The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Randomness,
More information2. 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 informationMULTIPLE 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 informationCombinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
More informationExamples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form
Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,
More information(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.
Examples for Chapter 3 Probability Math 10401 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw
More informationChapter 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