Probability COMP 245 STATISTICS. Dr N A Heard. 1 Sample Spaces and Events Sample Spaces Events Combinations of Events...
|
|
- Colleen Simmons
- 5 years ago
- Views:
Transcription
1 Probability COMP 245 STATISTICS Dr N A Heard Contents Sample Spaces and Events. Sample Spaces Events Combinations of Events Probability 3 2. Axioms of Probability Simple Probability Results Independence Interpretations of Probability Examples 3. De Méré s problem Joint events Conditional Probability 8 4. Definition Examples Conditional Independence Bayes Theorem and the Partition Rule More Examples Sample Spaces and Events. Sample Spaces We consider a random experiment whose range of possible outcomes can be described by a set S, called the sample space. We use S as our universal set (Ω). Ex. Coin tossing: S = {H, T}. Die rolling: S = {,,,,, }. 2 coins: S = {(H, H), (H, T), (T, H), (T, T)}.
2 .2 Events An event E is any subset of the sample space, E S; it is a collection of some of the possible outcomes. Ex. Coin tossing: E = {H}, E = {T}. Die rolling: E = { }, E = {Even numbered face} = {,, }. 2 coins: E = {Head on the first toss} = {(H, H), (H, T)}. Extreme possible events are (the null event) or S. The singleton subsets of S (those subsets which contain exactly one element from S) are known as the elementary events of S. Subsets of S containing more than one outcome are known as compound events. Suppose we now perform this random experiment; the outcome will be a single element s S. Then for any event E S, we will say E has occurred if and only if s E. If E has not occurred, it must be that s / E s E, so E has occurred; so E can be read as the event not E. First notice that the smallest event which will have occurred will be the singleton {s }. For any other event E, E will occur if and only if E is a compound event and {s } E. Thus we can immediately draw two conclusions before the experiment has even been performed. Remark.. For any sample space S, the following statements will always be true:. the null event will never occur; 2. the universal event S will always occur. Hence it is only for events E in between these extreme events, E S for which we have uncertainty about whether E will occur. It is precisely for quantifying this uncertainty over these events that we require the notion of probability..3 Combinations of Events Set operators on events Consider a set of events {E, E 2,...}. The event i E i = {s S i s.t. s E i } will occur if and only if at least one of the events {E i } occurs. So E E 2 can be read as event E or E 2. The event i E i = {s S i, s E i } will occur if and only if all of the events {E i } occur. So E E 2 can be read as events E and E 2. The events are said to be mutually exclusive if i, j, E i E j = (i.e. they are disjoint). At most one of the events can occur. 2
3 2 Probability 2. Axioms of Probability σ-algebras of events So for our random experiment with sample space S of possible outcomes, for which events/subsets E S would we like to define the probability of E occurring? Every subset? If S is finite or even countable, then this is fine. But for uncountably infinite sample spaces it can be shown that we can very easily start off defining sensible, proper probabilities for an initial collection of subsets of S in a way that leaves it impossible to then carry on and consistently define probability for all the remaining subsets of S. For this reason, when defining a probability measure on S we (usually implicitly) simultaneously agree on a collection of subsets of S that we wish to measure with probability. Generically, we will refer to this set of subsets as S. There are three properties we will require of S, the reasons for which will become immediately apparent when we meet the axioms of probability. We need S to be. nonempty, S S; 2. closed under complements: E S = E S; 3. closed under countable union E, E 2,... S = i E i S. Such a collection of sets is known as a σ-algebra. Axioms of Probability A probability measure on the pair (S, S) is a mapping P : S [0, ] satisfying the following three axioms for all subsets of S on which it is defined (S, the measurable subsets of S). E S, 0 P(E) ; 2. P(S) = ; 3. Countably additive: For disjoint subsets E, E 2,... S ( ) P E i = P(E i ). i i 2.2 Simple Probability Results Exercises: From -3 it is easy to derive the following:. P(E) = P(E); 2. P( ) = 0; 3. For any events E and F, P(E F) = P(E) + P(F) P(E F). 3
4 2.3 Independence Two events E and F are said to be independent if and only if P(E F) = P(E)P(F). This is sometimes written E F. More generally, a set of events {E, E 2,...} are said to be independent if for any finite subset {E, E 2,..., E n }, ( ) n n P E i = P(E i ). i= i= If events E and F are independent, then E and F are also independent. Proof: Since F = (E F) (E F) is a disjoint union, P(F) = P(E F) + P(E F) by Axiom 3. So P(E F) = P(F) P(E F) = P(F) P(E)P(F) = ( P(E))P(F) = P(E)P(F). 2.4 Interpretations of Probability Classical If S is finite and the elementary events are considered equally likely, then the probability of an event E is the proportion of all outcomes in S in which lie inside E, Ex. P(E) = E S. Rolling a die: Elementary events are { }, { },..., { }. P({ }) = P({ }) =... = P({ }) =. P(Odd number) = P({,, }) = 3 = 2. Randomly drawn playing card: 52 elementary events { 2}, { 3},..., { A}, { 2}, { 3},...,..., { K}, { A}. P( ) = P( ) = P( ) = P( ) = 4. The joint event {Suit is red and value is 3} contains two of 52 elementary events, so P({red 3}) = 2 52 = 2. Since suit and face value should be independent, check that P({red 3}) = P({, }) P({any 3}). The equally likely (uniform) idea can be extended to infinite spaces, by apportioning probability to sets not by their cardinality but by other standard measures, like volume or mass. Ex. If a meteorite were to strike Earth, the probability that it will strike land rather than sea would be given by Total area of land Total area of Earth. 4
5 Frequentist Observation shows that if one takes repeated observations in identical random situations, in which event E may or may not occur, then the proportion of times in which E occurs tends to some limiting value - called the probability of E Ex. Proportion of heads in tosses of a coin: H, H, T, H, T, T, H, T, T, Subjective Probability is a degree of belief held by an individual. For example, De Finetti (937/94) suggested the following: Suppose a random experiment is to be performed, where an event E S may or may not happen. Now suppose an individual is entered into a game regarding this experiment where he has two choices, each leading to monetary (or utility) consequences:. Gamble: If E occurs, he wins ; if E occurs, he wins 0; 2. Stick: Regardless of the outcome of the experiment, he receives P(E) for some real number P(E). The critical value of P(E) for which the individual is indifferent between options and 2 is defined to be the individual s probability for the event E occurring. This procedure can be repeated for all possible events E in S. Suppose after this process of elicitation of the individual s preferences under the different events, we can simultaneously arrange an arbitrary number of monetary bets with the individual based on the outcome of the experiment. If it is possible to choose these bets in such a way that the individual is certain to lose money (this is called a Dutch Book ), then the individuals degrees of belief are said to be incoherent. To be coherent, it is easily seen, for example, that we must have 0 P(E) for all events E, E F = P(E) P(F), etc. 5
6 3 Examples 3. De Méré s problem Antoine Gombaud, chevalier de Méré (07-84) posed to Pascal the following gambling problem: which of these two events is more likely?. E = {4 rolls of a die yield at least one }. 2. F = {24 rolls of two dice yield at least one pair of }. De Méré observed that E seemed to lead to a profitable even money bet whereas F did not. We calculate P(E) and P(F).. Each roll of the die is independent from the previous rolls, and so there are 4 equally likely outcomes. Of these, 5 4 show no s. So the probability of no showing is So P(E), the probability of at least one showing, is = There are 3 24 equally likely outcomes here. Of these, don t show a. So the probability of no is So P(F), the probability of at least one, is = Hence P(E) > 2 > P(F). 3.2 Joint events Coin and Die Consider tossing a coin and rolling a die. We would consider each of the 2 possible combinations of Head/Tail and die value as equally likely. So we can construct a probability table: H 2 T From this table we can calculate the probability of any event we might be interested in, simply by adding up the probabilities of all the elementary events it contains. For example, the event of getting a head on the coin {H} = {(H, ), (H, ),..., (H, )}
7 has probability P({H}) = P({(H, )}) + P({(H, )}) P({(H, )}) = = 2. Notice the two experiments satisfy our probability definition of independence, since for example P({(H, )}) = 2 = 2 = P({H}) P({ }). Coin and Two Dice A crooked die called a top has the same faces on opposite sides. Suppose we have two dice, one normal and one which is a top with opposite faces numbered,, or. Now suppose we first flip the coin. If it comes up heads, we roll the normal die; tails, and we roll the top. To calculate the probability table easily, we notice that this is equivalent to the previous game using one normal die except with the change after tails that a roll of a is relabelled as a,,. So we can just merge those probabilities in the tails row. H T The probabilities of the different outcomes of the dice change according to the outcome of the coin toss. And note, for example, P({(H, )}) = 2 = 24 = 2 = P({H}) P({ }). 2 So the two experiments are now dependent. 7
8 4 Conditional Probability 4. Definition For two events E and F in S where P(F) = 0, we define the conditional probability of E occurring given that we know F has occurred as P(E F) = P(E F). P(F) Note that if E and F are independent, then P(E F) = P(E F) P(F) = P(E)P(F) P(F) = P(E). 4.2 Examples Example - Rolling a We roll a normal die once.. What is the probability of E = {the die shows a }? 2. What is the probability of E = {the die shows a } given we know F = {the die shows an odd number}? Solution:. P(E) = Number of ways a can come up Total number of possible outcomes =. 2. Now the set of possible outcomes is just F = {,, }. Number of ways a can come up So P(E F) = Total number of possible outcomes = 3. Note P(F) = 2 P(E F) and E F = E, and hence we have P(E F) =. P(F) Example 2 - Rolling two dice Suppose we roll two normal dice, one from each hand. Then the sample space comprises all of the ordered pairs of dice values S = {(, ), (, ),..., (, )}. Let E be the event that the die thrown from the left hand will show a larger value than the die thrown from the right hand. P(E) = # outcomes with left value > right total # outcomes 8 = 5 3.
9 Suppose we are now informed that an event F has occurred, where F = {the value of the left hand die is } How does this change the probability of E occurring? Well since F has occurred, the only sample space elements which could have possibly occurred are exactly those elements in F = {(, ), (, ), (, ), (, ), (, )(, )}. Similarly the only sample space elements in E that could have occurred now must be in E F = {(, ), (, ), (, ), (, )}. So our revised probability is # outcomes with left value > right total # outcomes (, ) = 4 = P(E F) P(F) P(E F). Discussion of Examples In both examples, we considered the probability of an event E, and then reconsidered what this probability would be if we were given the knowledge that F had occurred. What happened? Answer: The sample space S was replaced by F, and the event E was replaced by E F. So originally, we had P(E) = P(E S) = P(E S) P(S) (since E S = E, and P(S) = by Axiom 2). So we can think of probability conditioning as a shrinking of the sample space, with events replaced by their intersections with the reduced space and a consequent rescaling of probabilities. 4.3 Conditional Independence Earlier we met the concept of independence of events according to a probability measure P. We can now extend that idea to conditional probabilities since P( F) is itself a perfectly good probability measure obeying the axioms of probability. For three events E, E 2 and F, the event pair E and E 2 are said to be conditionally independent given F if and only if P(E E 2 F) = P(E F)P(E 2 F). This is sometimes written E E 2 F. 4.4 Bayes Theorem and the Partition Rule Bayes Theorem For two events E and F in S, we have but also, since E F F E P(E F) = P(F)P(E F); () P(E F) = P(E)P(F E). (2) Equating the RHS of () and (2), provided P(F) = 0 we can rearrange to obtain P(E F) = P(E)P(F E). P(F) 9
10 Partition Rule Consider a set of events {F, F 2,...} which form a partition of S. Then for any event E S P(E) = P(E F i )P(F i ). i Proof: E = E S = E i F i = i (E F i ). So ( ) P(E) = P (E F i ), i which, by countable additivity (Axiom 3) and noting that since the {F, F 2,...} are disjoint so are {E F, E F 2,...}, implies P(E) = P(E F i ). (3) i (3) is known as the law of total probability; and it can be rewritten P(E) = P(E F i )P(F i ). i For any events E and F in S, note that {F, F}, say, form a partition of S. So by the law of total probability we have P(E) = P(E F) + P(E F) = P(E F)P(F) + P(E F)P(F). Terminology When considering multiple events, say E and F, we often refer to probabilities of the form P(E F) as conditional probabilities; probabilities of the form P(E F) as joint probabilities; probabilities of the form P(E) as marginal probabilities. 0
11 4.5 More Examples Example - Defective Chips Ex. A box contains 5000 VLSI chips, 000 from company X and 4000 from Y. 0% of the chips made by X are defective and 5% of those made by Y are defective. If a randomly chosen chip is found to be defective, find the probability that it came from X. Let E = chip was made by X ; let F = chip is defective. First of all, which probabilities have we been given? A box contains 5000 VLSI chips, 000 from company X and 4000 from Y. = P(E) = = 0.2, P(E) = = % of the chips made by X are defective and 5% of those made by Y are defective. = P(F E) = 0% = 0., P(F E) = 5% = We have enough information to construct the probability table E E F F The law of total probability has enabled us to extract the marginal probabilities P(F) and P(F) as 0.0 and 0.94 respectively. So by Bayes Theorem we can calculate the conditional probabilities. In particular, we want P(E F) = P(E F) P(F) = = 3.
12 Example 2 - Kidney stones Kidney stones are small (< 2cm diam) or large (> 2 cm diam). Treatment can succeed or fail. The following data were collected from a sample of 700 patients with kidney stones. Success (S) Failure (S) Large (L) Small (L) Total For a patient randomly drawn from this sample, what is the probability that the outcome of treatment was successful, given the kidney stones were large? Clearly we can get the answer directly from the table by ignoring the small stone patients or we can go the long way round: P(L) = , P(S L) = , P(S L) = P(S L) = P(S L) P(L) = = Example 3 - Multiple Choice Question A multiple choice question has c available choices. Let p be the probability that the student knows the right answer, and p that he does not. When he doesn t know, he chooses an answer at random. Given that the answer the student chooses is correct, what is the probability that the student knew the correct answer? Let A be the event that the question is answered correctly; let K be the event that the student knew the correct answer. Then we require P(K A). By Bayes Theorem P(K A) = P(A K)P(K) P(A) and we know P(A K) = and P(K) = p, so it remains to find P(A). By the partition rule, P(A) = P(A K)P(K) + P(A K)P(K) and since P(A K) =, this gives c Hence P(A) = p + ( p). c P(K A) = p p + p c = cp cp + p. Note: the larger c is, the greater the probability that the student knew the answer, given that they answered correctly. 2
13 Example 4 - Super Computer Jobs Measurements at the North Carolina Super Computing Center (NCSC) on a certain day showed that 5% of the jobs came from Duke, 35% from UNC, and 50% from NC State University. Suppose that the probabilities that each of these jobs is a multitasking job is 0.0, 0.05, and 0.02 respectively.. Find the probabilities that a job chosen at random is a multitasking job. 2. Find the probability that a randomly chosen job comes from UNC, given that it is a multitasking job. Solution: Let U i = job is from university i, i =, 2, 3 for Duke, UNC, NC State respectively; let M = job uses multitasking.. 2. P(M) = P(M U )P(U ) + P(M U 2 )P(U 2 ) + P(M U 3 )P(U 3 ) = = P(U 2 M) = P(M U 2)P(U 2 ) P(M) = = Example 5 - HIV Test A new HIV test is claimed to correctly identify 95% of people who are really HIV positive and 98% of people who are really HIV negative. Is this acceptable? If only in a 000 of the population are HIV positive, what is the probability that someone who tests positive actually has HIV? Solution: Let H = has the HIV virus ; let T = test is positive. We have been given P(T H) = 0.95, P(T H) = 0.98 and P(H) = We wish to find P(H T). P(T H)P(H) P(H T) = P(T H)P(H) + P(T H)P(H) = = That is, less than 5% of those who test positive really have HIV. 3
14 Example 5 - continued If the HIV test shows a positive result, the individual might wish to retake the test. Suppose that the results of a person retaking the HIV test are conditionally independent given HIV status (clearly two results of the test would certainly not be unconditionally independent). If the test again gives a positive result, what is the probability that the person actually has HIV? Solution: Let T i = i th test is positive. P(H T T 2 ) = P(T T 2 H)P(H) P(T T 2 ) P(T = T 2 H)P(H) P(T T 2 H)P(H) + P(T T 2 H)P(H) P(T = H)P(T 2 H)P(H) P(T H)P(T 2 H)P(H) + P(T H)P(T 2 H)P(H) by conditional independence. Since P(T i H) = 0.95 and P(T i H) = 0.02, P(H T T 2 ) = So almost a 70% chance after taking the test twice and both times showing as positive. For three times, this goes up to 99%. 4
Chapter 3 : Conditional Probability and Independence
STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 2016 Néhémy Lim Chapter 3 : Conditional Probability and Independence 1 Conditional Probabilities How should we modify the probability of an event when
More informationMATH 556: PROBABILITY PRIMER
MATH 6: PROBABILITY PRIMER 1 DEFINITIONS, TERMINOLOGY, NOTATION 1.1 EVENTS AND THE SAMPLE SPACE Definition 1.1 An experiment is a one-off or repeatable process or procedure for which (a there is a well-defined
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationProbability Year 10. Terminology
Probability Year 10 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationExample. What is the sample space for flipping a fair coin? Rolling a 6-sided die? Find the event E where E = {x x has exactly one head}
Chapter 7 Notes 1 (c) Epstein, 2013 CHAPTER 7: PROBABILITY 7.1: Experiments, Sample Spaces and Events Chapter 7 Notes 2 (c) Epstein, 2013 What is the sample space for flipping a fair coin three times?
More informationProbability Year 9. Terminology
Probability Year 9 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive
More informationLecture 1. Chapter 1. (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 ( ). 1. What is Statistics?
Lecture 1 (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 (2.1 --- 2.6). Chapter 1 1. What is Statistics? 2. Two definitions. (1). Population (2). Sample 3. The objective of statistics.
More informationI - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability
What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)
More informationFundamentals of Probability CE 311S
Fundamentals of Probability CE 311S OUTLINE Review Elementary set theory Probability fundamentals: outcomes, sample spaces, events Outline ELEMENTARY SET THEORY Basic probability concepts can be cast in
More informationTopic -2. Probability. Larson & Farber, Elementary Statistics: Picturing the World, 3e 1
Topic -2 Probability Larson & Farber, Elementary Statistics: Picturing the World, 3e 1 Probability Experiments Experiment : An experiment is an act that can be repeated under given condition. Rolling a
More informationUNIT 5 ~ Probability: What Are the Chances? 1
UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested
More informationthe time it takes until a radioactive substance undergoes a decay
1 Probabilities 1.1 Experiments with randomness Wewillusethetermexperimentinaverygeneralwaytorefertosomeprocess that produces a random outcome. Examples: (Ask class for some first) Here are some discrete
More informationSingle Maths B: Introduction to Probability
Single Maths B: Introduction to Probability Overview Lecturer Email Office Homework Webpage Dr Jonathan Cumming j.a.cumming@durham.ac.uk CM233 None! http://maths.dur.ac.uk/stats/people/jac/singleb/ 1 Introduction
More informationAxioms of Probability
Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing
More informationElementary Discrete Probability
Elementary Discrete Probability MATH 472 Financial Mathematics J Robert Buchanan 2018 Objectives In this lesson we will learn: the terminology of elementary probability, elementary rules of probability,
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14 Introduction One of the key properties of coin flips is independence: if you flip a fair coin ten times and get ten
More informationLecture 1: Probability Fundamentals
Lecture 1: Probability Fundamentals IB Paper 7: Probability and Statistics Carl Edward Rasmussen Department of Engineering, University of Cambridge January 22nd, 2008 Rasmussen (CUED) Lecture 1: Probability
More informationProblems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman.
Math 224 Fall 2017 Homework 1 Drew Armstrong Problems from Probability and Statistical Inference (9th ed.) by Hogg, Tanis and Zimmerman. Section 1.1, Exercises 4,5,6,7,9,12. Solutions to Book Problems.
More informationConditional Probability & Independence. Conditional Probabilities
Conditional Probability & Independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Definition: the conditional probability of E given F is P(E F
More informationChapter 2. Conditional Probability and Independence. 2.1 Conditional Probability
Chapter 2 Conditional Probability and Independence 2.1 Conditional Probability Example: Two dice are tossed. What is the probability that the sum is 8? This is an easy exercise: we have a sample space
More informationProbability 1 (MATH 11300) lecture slides
Probability 1 (MATH 11300) lecture slides Márton Balázs School of Mathematics University of Bristol Autumn, 2015 December 16, 2015 To know... http://www.maths.bris.ac.uk/ mb13434/prob1/ m.balazs@bristol.ac.uk
More informationStatistical Theory 1
Statistical Theory 1 Set Theory and Probability Paolo Bautista September 12, 2017 Set Theory We start by defining terms in Set Theory which will be used in the following sections. Definition 1 A set is
More informationSTT When trying to evaluate the likelihood of random events we are using following wording.
Introduction to Chapter 2. Probability. When trying to evaluate the likelihood of random events we are using following wording. Provide your own corresponding examples. Subjective probability. An individual
More informationOrigins of Probability Theory
1 16.584: INTRODUCTION Theory and Tools of Probability required to analyze and design systems subject to uncertain outcomes/unpredictability/randomness. Such systems more generally referred to as Experiments.
More informationConditional Probability
Conditional Probability When we obtain additional information about a probability experiment, we want to use the additional information to reassess the probabilities of events given the new information.
More informationLECTURE 1. 1 Introduction. 1.1 Sample spaces and events
LECTURE 1 1 Introduction The first part of our adventure is a highly selective review of probability theory, focusing especially on things that are most useful in statistics. 1.1 Sample spaces and events
More informationProbability Theory and Applications
Probability Theory and Applications Videos of the topics covered in this manual are available at the following links: Lesson 4 Probability I http://faculty.citadel.edu/silver/ba205/online course/lesson
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
More informationProbability. Chapter 1 Probability. A Simple Example. Sample Space and Probability. Sample Space and Event. Sample Space (Two Dice) Probability
Probability Chapter 1 Probability 1.1 asic Concepts 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
More informationRecap. The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY INFERENTIAL STATISTICS
Recap. Probability (section 1.1) The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY Population Sample INFERENTIAL STATISTICS Today. Formulation
More informationLecture notes for probability. Math 124
Lecture notes for probability Math 124 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result
More informationChapter 2. Conditional Probability and Independence. 2.1 Conditional Probability
Chapter 2 Conditional Probability and Independence 2.1 Conditional Probability Probability assigns a likelihood to results of experiments that have not yet been conducted. Suppose that the experiment has
More information2) There should be uncertainty as to which outcome will occur before the procedure takes place.
robability Numbers For many statisticians the concept of the probability that an event occurs is ultimately rooted in the interpretation of an event as an outcome of an experiment, others would interpret
More informationBASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES
BASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES COMMON TERMS RELATED TO PROBABILITY Probability is the measure of the likelihood that an event will occur Probability values are
More informationCompound Events. The event E = E c (the complement of E) is the event consisting of those outcomes which are not in E.
Compound Events Because we are using the framework of set theory to analyze probability, we can use unions, intersections and complements to break complex events into compositions of events for which it
More informationProperties of Probability
Econ 325 Notes on Probability 1 By Hiro Kasahara Properties of Probability In statistics, we consider random experiments, experiments for which the outcome is random, i.e., cannot be predicted with certainty.
More informationChapter Summary. 7.1 Discrete Probability 7.2 Probability Theory 7.3 Bayes Theorem 7.4 Expected value and Variance
Chapter 7 Chapter Summary 7.1 Discrete Probability 7.2 Probability Theory 7.3 Bayes Theorem 7.4 Expected value and Variance Section 7.1 Introduction Probability theory dates back to 1526 when the Italian
More informationLecture 3 Probability Basics
Lecture 3 Probability Basics Thais Paiva STA 111 - Summer 2013 Term II July 3, 2013 Lecture Plan 1 Definitions of probability 2 Rules of probability 3 Conditional probability What is Probability? Probability
More informationChapter 2: Probability Part 1
Engineering Probability & Statistics (AGE 1150) Chapter 2: Probability Part 1 Dr. O. Phillips Agboola Sample Space (S) Experiment: is some procedure (or process) that we do and it results in an outcome.
More informationP (A B) P ((B C) A) P (B A) = P (B A) + P (C A) P (A) = P (B A) + P (C A) = Q(A) + Q(B).
Lectures 7-8 jacques@ucsdedu 41 Conditional Probability Let (Ω, F, P ) be a probability space Suppose that we have prior information which leads us to conclude that an event A F occurs Based on this information,
More information(6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6)
Section 7.3: Compound Events Because we are using the framework of set theory to analyze probability, we can use unions, intersections and complements to break complex events into compositions of events
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #24: Probability Theory Based on materials developed by Dr. Adam Lee Not all events are equally likely
More informationIntroduction to Probability
Introduction to Probability Gambling at its core 16th century Cardano: Books on Games of Chance First systematic treatment of probability 17th century Chevalier de Mere posed a problem to his friend Pascal.
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationAnnouncements. Lecture 5: Probability. Dangling threads from last week: Mean vs. median. Dangling threads from last week: Sampling bias
Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure to
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 3 May 28th, 2009 Review We have discussed counting techniques in Chapter 1. Introduce the concept of the probability of an event. Compute probabilities in certain
More informationProbability Theory Review
Cogsci 118A: Natural Computation I Lecture 2 (01/07/10) Lecturer: Angela Yu Probability Theory Review Scribe: Joseph Schilz Lecture Summary 1. Set theory: terms and operators In this section, we provide
More informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationReview of Basic Probability Theory
Review of Basic Probability Theory James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 35 Review of Basic Probability Theory
More informationMATH MW Elementary Probability Course Notes Part I: Models and Counting
MATH 2030 3.00MW Elementary Probability Course Notes Part I: Models and Counting Tom Salisbury salt@yorku.ca York University Winter 2010 Introduction [Jan 5] Probability: the mathematics used for Statistics
More informationModule 1. Probability
Module 1 Probability 1. Introduction In our daily life we come across many processes whose nature cannot be predicted in advance. Such processes are referred to as random processes. The only way to derive
More informationDiscrete Probability
Discrete Probability Counting Permutations Combinations r- Combinations r- Combinations with repetition Allowed Pascal s Formula Binomial Theorem Conditional Probability Baye s Formula Independent Events
More informationProbability Theory. Introduction to Probability Theory. Principles of Counting Examples. Principles of Counting. Probability spaces.
Probability Theory To start out the course, we need to know something about statistics and probability Introduction to Probability Theory L645 Advanced NLP Autumn 2009 This is only an introduction; for
More informationMean, Median and Mode. Lecture 3 - Axioms of Probability. Where do they come from? Graphically. We start with a set of 21 numbers, Sta102 / BME102
Mean, Median and Mode Lecture 3 - Axioms of Probability Sta102 / BME102 Colin Rundel September 1, 2014 We start with a set of 21 numbers, ## [1] -2.2-1.6-1.0-0.5-0.4-0.3-0.2 0.1 0.1 0.2 0.4 ## [12] 0.4
More informationRandom processes. Lecture 17: Probability, Part 1. Probability. Law of large numbers
Random processes Lecture 17: Probability, Part 1 Statistics 10 Colin Rundel March 26, 2012 A random process is a situation in which we know what outcomes could happen, but we don t know which particular
More informationChapter 2 Class Notes
Chapter 2 Class Notes Probability can be thought of in many ways, for example as a relative frequency of a long series of trials (e.g. flips of a coin or die) Another approach is to let an expert (such
More informationProbability Experiments, Trials, Outcomes, Sample Spaces Example 1 Example 2
Probability 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. However, probability models underlie
More informationIntermediate Math Circles November 8, 2017 Probability II
Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B
More informationConditional Probability
Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the
More informationThe probability of an event is viewed as a numerical measure of the chance that the event will occur.
Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that
More information1 The Basic Counting Principles
1 The Basic Counting Principles The Multiplication Rule If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n ways [regardless of how
More informationCMPSCI 240: Reasoning about Uncertainty
CMPSCI 240: Reasoning about Uncertainty Lecture 2: Sets and Events Andrew McGregor University of Massachusetts Last Compiled: January 27, 2017 Outline 1 Recap 2 Experiments and Events 3 Probabilistic Models
More informationConditional Probability & Independence. Conditional Probabilities
Conditional Probability & Independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Definition: the conditional probability of E given F is P(E F
More informationLecture 3 - Axioms of Probability
Lecture 3 - Axioms of Probability Sta102 / BME102 January 25, 2016 Colin Rundel Axioms of Probability What does it mean to say that: The probability of flipping a coin and getting heads is 1/2? 3 What
More informationThe set of all outcomes or sample points is called the SAMPLE SPACE of the experiment.
Chapter 7 Probability 7.1 xperiments, Sample Spaces and vents Start with some definitions we will need in our study of probability. An XPRIMN is an activity with an observable result. ossing coins, rolling
More informationDynamic Programming Lecture #4
Dynamic Programming Lecture #4 Outline: Probability Review Probability space Conditional probability Total probability Bayes rule Independent events Conditional independence Mutual independence Probability
More informationIf S = {O 1, O 2,, O n }, where O i is the i th elementary outcome, and p i is the probability of the i th elementary outcome, then
1.1 Probabilities Def n: A random experiment is a process that, when performed, results in one and only one of many observations (or outcomes). The sample space S is the set of all elementary outcomes
More informationTopic 2 Probability. Basic probability Conditional probability and independence Bayes rule Basic reliability
Topic 2 Probability Basic probability Conditional probability and independence Bayes rule Basic reliability Random process: a process whose outcome can not be predicted with certainty Examples: rolling
More informationChapter 5 : Probability. Exercise Sheet. SHilal. 1 P a g e
1 P a g e experiment ( observing / measuring ) outcomes = results sample space = set of all outcomes events = subset of outcomes If we collect all outcomes we are forming a sample space If we collect some
More informationLecture Notes 1 Basic Probability. Elements of Probability. Conditional probability. Sequential Calculation of Probability
Lecture Notes 1 Basic Probability Set Theory Elements of Probability Conditional probability Sequential Calculation of Probability Total Probability and Bayes Rule Independence Counting EE 178/278A: Basic
More informationSTAT 516: Basic Probability and its Applications
Lecture 3: Conditional Probability and Independence Prof. Michael September 29, 2015 Motivating Example Experiment ξ consists of rolling a fair die twice; A = { the first roll is 6 } amd B = { the sum
More information1. When applied to an affected person, the test comes up positive in 90% of cases, and negative in 10% (these are called false negatives ).
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 8 Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According to clinical trials,
More informationStatistical Inference
Statistical Inference Lecture 1: Probability Theory MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 11, 2018 Outline Introduction Set Theory Basics of Probability Theory
More informationMath 1313 Experiments, Events and Sample Spaces
Math 1313 Experiments, Events and Sample Spaces At the end of this recording, you should be able to define and use the basic terminology used in defining experiments. Terminology The next main topic in
More informationChapter 1 (Basic Probability)
Chapter 1 (Basic Probability) What is probability? Consider the following experiments: 1. Count the number of arrival requests to a web server in a day. 2. Determine the execution time of a program. 3.
More information4. Probability of an event A for equally likely outcomes:
University of California, Los Angeles Department of Statistics Statistics 110A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment:
More informationIntroduction to probability
Introduction to probability 4.1 The Basics of Probability Probability The chance that a particular event will occur The probability value will be in the range 0 to 1 Experiment A process that produces
More informationProbability & Random Variables
& Random Variables Probability Probability theory is the branch of math that deals with random events, processes, and variables What does randomness mean to you? How would you define probability in your
More informationLecture Lecture 5
Lecture 4 --- Lecture 5 A. Basic Concepts (4.1-4.2) 1. Experiment: A process of observing a phenomenon that has variation in its outcome. Examples: (E1). Rolling a die, (E2). Drawing a card form a shuffled
More informationSTA Module 4 Probability Concepts. Rev.F08 1
STA 2023 Module 4 Probability Concepts Rev.F08 1 Learning Objectives Upon completing this module, you should be able to: 1. Compute probabilities for experiments having equally likely outcomes. 2. Interpret
More information3.2 Probability Rules
3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 1016-345-01: Probability and Statistics for Engineers Fall 2012 Contents 0 Administrata 2 0.1 Outline....................................... 3 1 Axiomatic Probability 3
More informationCIVL 7012/8012. Basic Laws and Axioms of Probability
CIVL 7012/8012 Basic Laws and Axioms of Probability Why are we studying probability and statistics? How can we quantify risks of decisions based on samples from a population? How should samples be selected
More informationHomework 4 Solution, due July 23
Homework 4 Solution, due July 23 Random Variables Problem 1. Let X be the random number on a die: from 1 to. (i) What is the distribution of X? (ii) Calculate EX. (iii) Calculate EX 2. (iv) Calculate Var
More informationProbabilistic models
Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became the definitive formulation
More informationProbability Theory and Random Variables
Probability Theory and Random Variables One of the most noticeable aspects of many computer science related phenomena is the lack of certainty. When a job is submitted to a batch oriented computer system,
More informationIntroduction to Probability
Introduction to Probability Content Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes Theorem 2
More informationCS626 Data Analysis and Simulation
CS626 Data Analysis and Simulation Instructor: Peter Kemper R 104A, phone 221-3462, email:kemper@cs.wm.edu Today: Probability Primer Quick Reference: Sheldon Ross: Introduction to Probability Models 9th
More informationLECTURE NOTES by DR. J.S.V.R. KRISHNA PRASAD
.0 Introduction: The theory of probability has its origin in the games of chance related to gambling such as tossing of a coin, throwing of a die, drawing cards from a pack of cards etc. Jerame Cardon,
More informationBrief Review of Probability
Brief Review of Probability Nuno Vasconcelos (Ken Kreutz-Delgado) ECE Department, UCSD Probability Probability theory is a mathematical language to deal with processes or experiments that are non-deterministic
More informationF71SM STATISTICAL METHODS
F71SM STATISTICAL METHODS RJG SUMMARY NOTES 2 PROBABILITY 2.1 Introduction A random experiment is an experiment which is repeatable under identical conditions, and for which, at each repetition, the outcome
More informationSTAT:5100 (22S:193) Statistical Inference I
STAT:5100 (22S:193) Statistical Inference I Week 3 Luke Tierney University of Iowa Fall 2015 Luke Tierney (U Iowa) STAT:5100 (22S:193) Statistical Inference I Fall 2015 1 Recap Matching problem Generalized
More informationConditional probability
CHAPTER 4 Conditional probability 4.1. Introduction Suppose there are 200 men, of which 100 are smokers, and 100 women, of which 20 are smokers. What is the probability that a person chosen at random will
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More informationIntroduction to Probability. Ariel Yadin. Lecture 1. We begin with an example [this is known as Bertrand s paradox]. *** Nov.
Introduction to Probability Ariel Yadin Lecture 1 1. Example: Bertrand s Paradox We begin with an example [this is known as Bertrand s paradox]. *** Nov. 1 *** Question 1.1. Consider a circle of radius
More informationBasic notions of probability theory
Basic notions of probability theory Contents o Boolean Logic o Definitions of probability o Probability laws Objectives of This Lecture What do we intend for probability in the context of RAM and risk
More informationSenior Math Circles November 19, 2008 Probability II
University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 9, 2008 Probability II Probability Counting There are many situations where
More informationPresentation on Theo e ry r y o f P r P o r bab a il i i l t i y
Presentation on Theory of Probability Meaning of Probability: Chance of occurrence of any event In practical life we come across situation where the result are uncertain Theory of probability was originated
More information4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio
4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio Wrong is right. Thelonious Monk 4.1 Three Definitions of
More information