Beginnings of Probability I


 Amy Terry
 2 years ago
 Views:
Transcription
1 Beginnings of Probability I Despite the fact that humans have played games of chance forever (so to speak), it is only in the 17 th century that two mathematicians, Pierre Fermat and Blaise Pascal, set up the foundations of probability. The area has undergone great development since, and is used to make predictions about natural phenomena. If you are involved in any science including finance and economics, you will run across probability and statistics. Example Pierre and Blaise are playing a game. They each contributed 50FF to a common pool. A coin is tossed. if heads comes up, Pierre wins; if tails come up, Blaise wins. The game ends whenever one reaches 10; he gets the 100FF. After 15 tries, Pierre has 8 and Blaise has 7. They have to stop the game for some reason. Question. How should they split the pot? You can read about the history and reasoning for this type of problem at of points Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 1 / 19
2 PROBABILITY I Sample Space (Same as/analogue of Universal Set) Experiment an activity/occurence with an observable result. If it repeats, call it a trial. Event Subset of a Sample Space. Mutually Exclusive Events A pair of disjoints sets, A, B such that A B =. Basic Probability. The majority of our sample spaces S are finite. We associate numbers to each event p(a) such that: 0 p(a) 1, p(s) = 1, If A, B are mutually exclusive, p(a B) = p(a) + p(b). So if S = {a 1,..., a n } it is enough to associate a number 0 p({a i }) = p i 1 to each element so that the sum is equal to 1. The {p i } are called a Probability Distribution. Uniform Distribution. p i = 1 n. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 2 / 19
3 Examples YOUR TURN 2. Two fair coins are tossed. Record the outcomes and the probabilities. Fair means uniform distribution. The probabilities of each outcome is the same. S = {HH, HT, TH, TT } The probability of each singleton outcome is 1 4. The probability of one head and one tail is A = {HT, TH}, p(a) = p({ht }) + p({th}) = = 1 2. The probability of at least one head: A = {HH, HT, TH}, p(a) = n(a) n(s) = 3 4. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 3 / 19
4 Empirical Probability I In reality we are never given the probability distribution. We guess what it is from a sample of data, and use it to make predictions about similar instances. This is called Empirical Probability in the text. It it closely related to Statistics. 57. Causes of Death. There were 2,424,059 U.S. deaths in They are listed according to cause in the following table. If a randomly selected person died in 2007, use this information to find the following probabilities. Source: Centers for Disease Control and Prevention. If a randomly selected person died in 2007, use this information to find the following probabilities. Source: Centers for Disease Control and Prevention. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 4 / 19
5 Cause Number of Deaths Heart Disease 615,651 Cancer 560,187 Cerebrovascular disease 133,990 Chronic lower respiratory disease 129,311 Accidents 117,075 Alzheimer s disease 74,944 Diabetes mellitus 70,905 Influenza and pneumonia 52,847 All other causes 669,149 a. The probability that the cause of death was heart disease. b. The probability that the cause of death was cancer or heart disease. a. The probability that the cause of death was heart disease b. The probability that the cause of death was cancer or heart Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 5 / 19
6 c. The probability that the cause of death was not an accident and was not diabetes mellitus. ANSWER. We assume uniform distribution. The sample space has size n(s) =615, , , , , , 944+ a) 615, b) 615, , , , 149 = c) Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 6 / 19
7 Preliminary Questions about the Problem of Points 1 How do we set up the game in terms of the notions on the first slide? I mean sample space, event, trial... 2 What are the assumptions about the coin toss? 3 How long will the game last? Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 7 / 19
8 Answers I The sample space is {H, T }. The experiment is to toss the coin and record the outcome. {H} and {T } are mutually exclusive events. We repeat the experiment, so the whole game would be the trial. We can also set it up differently. There will be at most 19 tosses. Record all possible outcomes. That would be the sample space, and an event would be any subset. We are interested in the singletons, sets with just one element, one outcome of the game. This would be very awkward to write out. Either way, we have to associate a probability distribution. For this we need some assumtions. The following are reasonable: The coin is fair. In any one given toss, it is equally likey that heads or tails appear: p(h) = 1/2, p(t ) = 1/2. The individual tosses are identical, and independent of each other. No outcome of a toss affects any other. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 8 / 19
9 Answers II We can now analyze the game. From 8 and 7 on, the following outcomes are possible. We record four tosses, each equally likely with probability 1/16. F HHHH HHHT HHTH HTHH THHH HHTT HTHT HTTH F THHT THTH TTHH B HTTT THTT TTHT TTTH TTTT There are 11 ways F wins, 5 ways P wins. So they decide to split the pot Instead of writing HHHH, we could just write HH because the game stops. But then we would assign probability 1 4 to this event. F HH HTH THH HTTH TTHH THTH 1/4 1/8 1/8 1/16 1/16 1/16 11/16 B HTTT TTT THTT TTHT 1/16 1/8 1/16 1/16 5/16 Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August 27 9 / 19
10 More Questions What is the probablity that the game will end in 1 One toss 2 Two tosses 3 Three tosses 4 Four tosses 5 Five or more tosses Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
11 Probability Rules I Union Rule. P(A B) = P(A) + P(B) P(A B). Complement Rule. P(A c ) = 1 P(A). Odds If the odds of E occuring are m to n, then p(e) = p(e c ) =. n should not be 0. n m+n m m+n and Conditional Probability. P(A B) = P(A B) P(B). Careful about P(B) = 0. Product Rule p(a B) = p(b) p(a B) = p(a) p(b A). Independent Events. P(A B) = P(A) P(B) same as P(A B) = P(A) same as P(B A) = P(B). Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
12 A standard deck of 52 cards has four suits: hearts, clubs, diamonds, and spades, with 13 cards in each suit. The hearts and diamonds are red, and the spades and clubs are black. Each suit has an ace (A), a king (K), a queen (Q), a jack (J), and cards numbered from 2 to 10. The jack, queen, and king are called face cards and for many purposes can be thought of as having values 11, 12, and 13, respectively. The ace can be thought of as the low card (value 1) or the high card (value 14). See Figure 17. We will refer to this standard deck of cards often in our discussion of probability. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
13 can be thought of as having values 11, 12, and 13, respectively. The ace can be thought of as the low card (value 1) or the high card (value 14). See Figure 17. We will refer to this standard deck of cards often in our discussion of probability. Face cards Aces Jacks Queens Kings Hearts Clubs Diamonds Spades FIGURE 17 EXAMPLE 7 Playing Cards If a single playing card is drawn at random from a standard 52card deck, find the probability of each event. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
14 Banking The Midtown Bank has found that most customers at the tellers windows either cash a check or make a deposit. Banking The following The Midtown table indicates Bank has found the transactions that most customers for one at teller the tellers for windows one day. either cash a check or make a deposit. The following table indicates the transactions for one teller for one day. Cash Check No Check Totals Conditional Probability I Make Deposit No Deposit Totals Letting Letting C represent represent cashing cashing a check a check and Dand represent D represent making deposit making express a deposit, each express probability each in words probability and find in its words value. and nd its value. 36. P1 C 1 D P1 Dr 1 C 2 p(c D) p(d c C) 38. P1 Cr 1 Dr 2 p(c c D c ) 39. P1 Cr 1 D 2 p(c40. c D) P31C p(c > D D) 2 r4 p((c D) c ) 41. Airline Delays In February 2010, the major U.S. airline with the fewest delays was United Airlines, for which 77.3% of their Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
15 Examples I Example (1) Two fair dice are thrown. What is the probability that a three shows or that the sum is larger than three? Example (2) A card is drawn from a wellshuffled deck. What is the probability that it is not a face card? Example (3) If the odds in favor of a horse s winning are 4 to 6, what is the probability that the horse will win the race? Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
16 Examples II Example (Example 7 in Section 7.5, YOUR TURN 6) The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and 1 in New York. Due to a lack of inspectors, they decide to inspect two plants selected at random, one this month and one next month, with each plant equally likely to be selected, but no plant selected twice. What is the probability that 1 New York plant and 1 Chicago plant are selected? Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
17 Independent Events 48. Medical Experiment A medical experiment showed that the probability that a new medicine is effective is 0.75, the probability that a patient will have a certain side effect is 0.4, and the probability that both events occur is 0.3. Decide whether these events are dependent or independent. Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
18 Drawing Cards without Replacement I Example Draw two cards from a well shuffled deck without replacement. Are the two events A = { first card is an ace } independent? B = { second card is a heart } Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
19 Drawing Cards without Replacement II Answer. For a tree diagram check your class notes. A well shuffled deck means/implies uniform distribution, any pair of two cards is equally likely to be drawn. There are equally likely choices. For the first card there are four choices of ace, spade, club, heart and diamond. For each choice of a first card there are 51 choices of a second card. So there are 4 51 choices for the second card. Then p(a) = = For B, there are 52 choices of the first card. Thirteen are hearts and 39 are other. For the 13 choices of hearts there are 12 choices of hearts for the second card. For the other 39 there are 13 choices of hearts for the second card. So p(b) = For A B there are four choices of aces for the first card. For the choice of ace of hearts there are 12 choices of hearts for the second card. For the other three aces there are 13 choices of hearts for the second card. So p(a B) = You can check that p(a B) p(a) p(b). Dan Barbasch Math 1105 Chapter 7.3, 7.4, 7.5 Week of August / 19
HUDM4122 Probability and Statistical Inference. February 2, 2015
HUDM4122 Probability and Statistical Inference February 2, 2015 In the last class Covariance Correlation Scatterplots Simple linear regression Questions? Comments? Today Ch. 4.14.3 in Mendenhall, Beaver,
More informationClass XII Chapter 13 Probability Maths. Exercise 13.1
Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 12: More Probability Tessa L. ChildersDay UC Berkeley 10 July 2014 By the end of this lecture... You will be able to: Use the theory of equally likely
More informationExercise Class XI Chapter 16 Probability Maths
Exercise 16.1 Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In daytoday life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting
More information= = 0.1%. On the other hand, if there are three winning tickets, then the probability of winning one of these winning tickets must be 3 (1)
MA 5 Lecture  Binomial Probabilities Wednesday, April 25, 202. Objectives: Introduce combinations and Pascal s triangle. The Fibonacci sequence had a number pattern that we could analyze in different
More informationOutcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}
Section 8: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as
More informationRandom Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }
Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,
More informationRandom Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment.
Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,
More informationCounting methods (Part 4): More combinations
April 13, 2009 Counting methods (Part 4): More combinations page 1 Counting methods (Part 4): More combinations Recap of last lesson: The combination number n C r is the answer to this counting question:
More informationMATH , Summer I Homework  05
MATH 230002, Summer I  200 Homework  05 Name... TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Due on Tuesday, October 26th ) True or False: If p remains constant
More information3. a. P(white) =, or. b. ; the probability of choosing a white block. d. P(white) =, or. 4. a. = 1 b. 0 c. = 0
Answers Investigation ACE Assignment Choices Problem. Core, 6 Other Connections, Extensions Problem. Core 6 Other Connections 7 ; unassigned choices from previous problems Problem. Core 7 9 Other Connections
More informationMath 1313 Section 6.2 Definition of Probability
Math 1313 Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability
More informationProbability. Ms. Weinstein Probability & Statistics
Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random
More informationb. 2 ; the probability of choosing a white d. P(white) 25, or a a. Since the probability of choosing a
Applications. a. P(green) =, P(yellow) = 2, or 2, P(red) = 2 ; three of the four blocks are not red. d. 2. a. P(green) = 2 25, P(purple) = 6 25, P(orange) = 2 25, P(yellow) = 5 25, or 5 2 6 2 5 25 25 25
More informationIntermediate Math Circles November 1, 2017 Probability I
Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.
More information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationSTAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes
STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationGrade 6 Math Circles Fall Oct 14/15 Probability
1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014  Oct 14/15 Probability Probability is the likelihood of an event occurring.
More informationSimple Probability. Arthur White. 28th September 2016
Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing
More informationThe probability setup
CHAPTER 2 The probability setup 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample
More informationThe probability setup
CHAPTER The probability setup.1. Introduction and basic theory We will have a sample space, denoted S sometimes Ω that consists of all possible outcomes. For example, if we roll two dice, the sample space
More informationProbability Rules. 2) The probability, P, of any event ranges from which of the following?
Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,
More informationSTATISTICS and PROBABILITY GRADE 6
Kansas City Area Teachers of Mathematics 2015 KCATM Math Competition STATISTICS and PROBABILITY GRADE 6 INSTRUCTIONS Do not open this booklet until instructed to do so. Time limit: 20 minutes You may use
More informationCHAPTERS 14 & 15 PROBABILITY STAT 203
CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical
More informationSection Introduction to Sets
Section 1.1  Introduction to Sets Definition: A set is a welldefined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationExam 2 Review F09 O Brien. Finite Mathematics Exam 2 Review
Finite Mathematics Exam Review Approximately 5 0% of the questions on Exam will come from Chapters, 4, and 5. The remaining 70 75% will come from Chapter 7. To help you prepare for the first part of the
More informationPROBABILITY. 1. Introduction. Candidates should able to:
PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation
More informationMath Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.
Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166  Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1  Experiments, Sample Spaces,
More informationMath Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.
Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166  Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1  Experiments, Sample Spaces,
More informationSuch a description is the basis for a probability model. Here is the basic vocabulary we use.
5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these
More informationChapter 5: Probability: What are the Chances? Section 5.2 Probability Rules
+ Chapter 5: Probability: What are the Chances? Section 5.2 + TwoWay Tables and Probability When finding probabilities involving two events, a twoway table can display the sample space in a way that
More informationWhen combined events A and B are independent:
A Resource for reestanding Mathematics Qualifications A or B Mutually exclusive means that A and B cannot both happen at the same time. Venn Diagram showing mutually exclusive events: Aces The events
More informationUNIT 4 APPLICATIONS OF PROBABILITY Lesson 1: Events. Instruction. Guided Practice Example 1
Guided Practice Example 1 Bobbi tosses a coin 3 times. What is the probability that she gets exactly 2 heads? Write your answer as a fraction, as a decimal, and as a percent. Sample space = {HHH, HHT,
More informationProbability. Dr. Zhang Fordham Univ.
Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!
More informationApplications of Probability
Applications of Probability CK12 Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive
More information4.3 Rules of Probability
4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:
More informationIndependent and Mutually Exclusive Events
Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationChapter 1: Sets and Probability
Chapter 1: Sets and Probability Section 1.31.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes
More informationFdaytalk.com. Outcomes is probable results related to an experiment
EXPERIMENT: Experiment is Definite/Countable probable results Example: Tossing a coin Throwing a dice OUTCOMES: Outcomes is probable results related to an experiment Example: H, T Coin 1, 2, 3, 4, 5, 6
More informationDef: The intersection of A and B is the set of all elements common to both set A and set B
Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:
More informationProbability  Chapter 4
Probability  Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person
More informationCopyright 2015 Edmentum  All rights reserved Picture is not drawn to scale.
Study Island Copyright 2015 Edmentum  All rights reserved. Generation Date: 05/26/2015 Generated By: Matthew Beyranevand Students Entering Grade 8 Part 2 Questions and Answers Compute with Rational Numbers
More information, the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.
41 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,
More informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 34 20142015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
More informationCHAPTER 7 Probability
CHAPTER 7 Probability 7.1. Sets A set is a welldefined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can
More informationBefore giving a formal definition of probability, we explain some terms related to probability.
probability 22 INTRODUCTION In our daytoday life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely
More informationIntroduction to probability
Introduction to probability Suppose an experiment has a finite set X = {x 1,x 2,...,x n } of n possible outcomes. Each time the experiment is performed exactly one on the n outcomes happens. Assign each
More informationThe study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical
More information05 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.
1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. d. a. Copy the table and add a column to show the experimental probability of the spinner landing on
More informationProbability: Terminology and Examples Spring January 1, / 22
Probability: Terminology and Examples 18.05 Spring 2014 January 1, 2017 1 / 22 Board Question Deck of 52 cards 13 ranks: 2, 3,..., 9, 10, J, Q, K, A 4 suits:,,,, Poker hands Consists of 5 cards A onepair
More informationProbability and Statistics. Copyright Cengage Learning. All rights reserved.
Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.2 Probability Copyright Cengage Learning. All rights reserved. Objectives What Is Probability? Calculating Probability by
More informationIf you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics
If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements
More informationFALL 2012 MATH 1324 REVIEW EXAM 4
FALL 01 MATH 134 REVIEW EXAM 4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the sample space for the given experiment. 1) An ordinary die
More informationProbability. Probabilty Impossibe Unlikely Equally Likely Likely Certain
PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0
More informationDiamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES
CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times
More informationSTAT 155 Introductory Statistics. Lecture 11: Randomness and Probability Model
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 11: Randomness and Probability Model 10/5/06 Lecture 11 1 The Monty Hall Problem Let s Make A Deal: a game show
More informationSection 6.5 Conditional Probability
Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability
More informationSTOR 155 Introductory Statistics. Lecture 10: Randomness and Probability Model
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model 10/6/09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal: a game show
More informationLesson 4: Chapter 4 Sections 12
Lesson 4: Chapter 4 Sections 12 Caleb Moxley BSC Mathematics 14 September 15 4.1 Randomness What s randomness? 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More information4.2.4 What if both events happen?
4.2.4 What if both events happen? Unions, Intersections, and Complements In the mid 1600 s, a French nobleman, the Chevalier de Mere, was wondering why he was losing money on a bet that he thought was
More informationProbability Models. Section 6.2
Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example
More informationKey Concepts. Theoretical Probability. Terminology. Lesson 111
Key Concepts Theoretical Probability Lesson  Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationCHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events
CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More informationDefine and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the
More informationChapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance FreeResponse 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is
More informationCounting and Probability
0838 ch0_p639693 0//007 0:3 PM Page 633 CHAPTER 0 Counting and Probability The design below is like a seed puff of a dandelion just before it is dispersed by the wind. The design shows the outcomes from
More informationGrade 7/8 Math Circles February 25/26, Probability
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely
More information1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
More informationABC High School, Kathmandu, Nepal. Topic : Probability
BC High School, athmandu, Nepal Topic : Probability Grade 0 Teacher: Shyam Prasad charya. Objective of the Module: t the end of this lesson, students will be able to define and say formula of. define Mutually
More informationClassical vs. Empirical Probability Activity
Name: Date: Hour : Classical vs. Empirical Probability Activity (100 Formative Points) For this activity, you will be taking part in 5 different probability experiments: Rolling dice, drawing cards, drawing
More 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 information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
More information1. The chance of getting a flush in a 5card poker hand is about 2 in 1000.
CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 15 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette wheels. Today
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 6 Lecture Notes Discrete Probability Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. Introduction and
More informationProbability Exercise 2
Probability Exercise 2 1 Question 9 A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will
More informationObjective: Determine empirical probability based on specific sample data. (AA21)
Do Now: What is an experiment? List some experiments. What types of things does one take a "chance" on? Mar 1 3:33 PM Date: Probability  Empirical  By Experiment Objective: Determine empirical probability
More informationAxiomatic Probability
Axiomatic Probability The objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance thtat A will occur.
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, InclusionExclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More informationProbability as a general concept can be defined as the chance of an event occurring.
3. Probability In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. Probability as a general
More informationExam III Review Problems
c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous WeekinReviews
More informationWeek 3 Classical Probability, Part I
Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability
More informationChapter 3: PROBABILITY
Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of
More informationPROBABILITY Case of cards
WORKSHEET NO1 PROBABILITY Case of cards WORKSHEET NO2 Case of two die Case of coins WORKSHEET NO3 1) Fill in the blanks: A. The probability of an impossible event is B. The probability of a sure
More informationThe topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
More informationChapter 5  Elementary Probability Theory
Chapter 5  Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling
More informationTotal. STAT/MATH 394 A  Autumn Quarter Midterm. Name: Student ID Number: Directions. Complete all questions.
STAT/MATH 9 A  Autumn Quarter 015  Midterm Name: Student ID Number: Problem 1 5 Total Points Directions. Complete all questions. You may use a scientific calculator during this examination; graphing
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is
More information