7.1 Sample Space, Events, Probability
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Descriptive Statistics (Part 2): Interpreting Study Results
Statistical Notes II Descriptive statistics (Part 2): Interpreting study results A Cook and A Sheikh he previous paper in this series looked at ‘baseline’. Investigations of treatment effects can be descriptive statistics, showing how to use and made in similar fashion by comparisons of disease T interpret fundamental measures such as the probability in treated and untreated patients. mean and standard deviation. Here we continue with descriptive statistics, looking at measures more The relative risk (RR), also sometimes known as specific to medical research. We start by defining the risk ratio, compares the risk of exposed and risk and odds, the two basic measures of disease unexposed subjects, while the odds ratio (OR) probability. Then we show how the effect of a disease compares odds. A relative risk or odds ratio greater risk factor, or a treatment, can be measured using the than one indicates an exposure to be harmful, while relative risk or the odds ratio. Finally we discuss the a value less than one indicates a protective effect. ‘number needed to treat’, a measure derived from the RR = 1.2 means exposed people are 20% more likely relative risk, which has gained popularity because of to be diseased, RR = 1.4 means 40% more likely. its clinical usefulness. Examples from the literature OR = 1.2 means that the odds of disease is 20% higher are used to illustrate important concepts. in exposed people. RISK AND ODDS Among workers at factory two (‘exposed’ workers) The probability of an individual becoming diseased the risk is 13 / 116 = 0.11, compared to an ‘unexposed’ is the risk. -
Teacher Guide 12.1 Gambling Page 1
TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial Literacy Objective 12.1: Analyze the probabilities involved in winning at games of chance. Objective 12.2: Evaluate costs and benefits of gambling to Simone, Paula, and individuals and society (e.g., family budget; addictive behaviors; Randy meet in the library and the local and state economy). every afternoon to work on their homework. Here is what is going on. Ryan is flipping a coin, and he is not cheating. He has just flipped seven heads in a row. Is Ryan’s next flip more likely to be: Heads; Tails; or Heads and tails are equally likely. Paula says heads because the first seven were heads, so the next one will probably be heads too. Randy says tails. The first seven were heads, so the next one must be tails. Lesson Objectives Simone says that it could be either heads or tails Recognize gambling as a form of risk. because both are equally Calculate the probabilities of winning in games of chance. likely. Who is correct? Explore the potential benefits of gambling for society. Explore the potential costs of gambling for society. Evaluate the personal costs and benefits of gambling. © 2008. Oklahoma State Department of Education. All rights reserved. Teacher Guide 12.1 2 Personal Financial Literacy Vocabulary Dependent event: The outcome of one event affects the outcome of another, changing the TEACHING TIP probability of the second event. This lesson is based on risk. -
Random Variable = a Real-Valued Function of an Outcome X = F(Outcome)
Random Variables (Chapter 2) Random variable = A real-valued function of an outcome X = f(outcome) Domain of X: Sample space of the experiment. Ex: Consider an experiment consisting of 3 Bernoulli trials. Bernoulli trial = Only two possible outcomes – success (S) or failure (F). • “IF” statement: if … then “S” else “F” • Examine each component. S = “acceptable”, F = “defective”. • Transmit binary digits through a communication channel. S = “digit received correctly”, F = “digit received incorrectly”. Suppose the trials are independent and each trial has a probability ½ of success. X = # successes observed in the experiment. Possible values: Outcome Value of X (SSS) (SSF) (SFS) … … (FFF) Random variable: • Assigns a real number to each outcome in S. • Denoted by X, Y, Z, etc., and its values by x, y, z, etc. • Its value depends on chance. • Its value becomes available once the experiment is completed and the outcome is known. • Probabilities of its values are determined by the probabilities of the outcomes in the sample space. Probability distribution of X = A table, formula or a graph that summarizes how the total probability of one is distributed over all the possible values of X. In the Bernoulli trials example, what is the distribution of X? 1 Two types of random variables: Discrete rv = Takes finite or countable number of values • Number of jobs in a queue • Number of errors • Number of successes, etc. Continuous rv = Takes all values in an interval – i.e., it has uncountable number of values. • Execution time • Waiting time • Miles per gallon • Distance traveled, etc. Discrete random variables X = A discrete rv. -
Odds: Gambling, Law and Strategy in the European Union Anastasios Kaburakis* and Ryan M Rodenberg†
63 Odds: Gambling, Law and Strategy in the European Union Anastasios Kaburakis* and Ryan M Rodenberg† Contemporary business law contributions argue that legal knowledge or ‘legal astuteness’ can lead to a sustainable competitive advantage.1 Past theses and treatises have led more academic research to endeavour the confluence between law and strategy.2 European scholars have engaged in the Proactive Law Movement, recently adopted and incorporated into policy by the European Commission.3 As such, the ‘many futures of legal strategy’ provide * Dr Anastasios Kaburakis is an assistant professor at the John Cook School of Business, Saint Louis University, teaching courses in strategic management, sports business and international comparative gaming law. He holds MS and PhD degrees from Indiana University-Bloomington and a law degree from Aristotle University in Thessaloniki, Greece. Prior to academia, he practised law in Greece and coached basketball at the professional club and national team level. He currently consults international sport federations, as well as gaming operators on regulatory matters and policy compliance strategies. † Dr Ryan Rodenberg is an assistant professor at Florida State University where he teaches sports law analytics. He earned his JD from the University of Washington-Seattle and PhD from Indiana University-Bloomington. Prior to academia, he worked at Octagon, Nike and the ATP World Tour. 1 See C Bagley, ‘What’s Law Got to Do With It?: Integrating Law and Strategy’ (2010) 47 American Business Law Journal 587; C -
3. Probability Theory
Ismor Fischer, 5/29/2012 3.1-1 3. Probability Theory 3.1 Basic Ideas, Definitions, and Properties POPULATION = Unlimited supply of five types of fruit, in equal proportions. O1 = Macintosh apple O4 = Cavendish (supermarket) banana O2 = Golden Delicious apple O5 = Plantain banana O3 = Granny Smith apple … … … … … … Experiment 1: Randomly select one fruit from this population, and record its type. Sample Space: The set S of all possible elementary outcomes of an experiment. S = {O1, O2, O3, O4, O5} #(S) = 5 Event: Any subset of a sample space S. (“Elementary outcomes” = simple events.) A = “Select an apple.” = {O1, O2, O3} #(A) = 3 B = “Select a banana.” = {O , O } #(B) = 2 #(Event) 4 5 #(trials) 1/1 1 3/4 2/3 4/6 . Event P(Event) A 3/5 0.6 1/2 A 3/5 = 0.6 0.4 B 2/5 . 1/3 2/6 1/4 B 2/5 = 0.4 # trials of 0 experiment 1 2 3 4 5 6 . 5/5 = 1.0 e.g., A B A A B A . P(A) = 0.6 “The probability of randomly selecting an apple is 0.6.” As # trials → ∞ P(B) = 0.4 “The probability of randomly selecting a banana is 0.4.” Ismor Fischer, 5/29/2012 3.1-2 General formulation may be facilitated with the use of a Venn diagram: Experiment ⇒ Sample Space: S = {O1, O2, …, Ok} #(S) = k A Om+1 Om+2 O2 O1 O3 Om+3 O 4 . Om . Ok Event A = {O1, O2, …, Om} ⊆ S #(A) = m ≤ k Definition: The probability of event A, denoted P(A), is the long-run relative frequency with which A is expected to occur, as the experiment is repeated indefinitely. -
Probability Cheatsheet V2.0 Thinking Conditionally Law of Total Probability (LOTP)
Probability Cheatsheet v2.0 Thinking Conditionally Law of Total Probability (LOTP) Let B1;B2;B3; :::Bn be a partition of the sample space (i.e., they are Compiled by William Chen (http://wzchen.com) and Joe Blitzstein, Independence disjoint and their union is the entire sample space). with contributions from Sebastian Chiu, Yuan Jiang, Yuqi Hou, and Independent Events A and B are independent if knowing whether P (A) = P (AjB )P (B ) + P (AjB )P (B ) + ··· + P (AjB )P (B ) Jessy Hwang. Material based on Joe Blitzstein's (@stat110) lectures 1 1 2 2 n n A occurred gives no information about whether B occurred. More (http://stat110.net) and Blitzstein/Hwang's Introduction to P (A) = P (A \ B1) + P (A \ B2) + ··· + P (A \ Bn) formally, A and B (which have nonzero probability) are independent if Probability textbook (http://bit.ly/introprobability). Licensed and only if one of the following equivalent statements holds: For LOTP with extra conditioning, just add in another event C! under CC BY-NC-SA 4.0. Please share comments, suggestions, and errors at http://github.com/wzchen/probability_cheatsheet. P (A \ B) = P (A)P (B) P (AjC) = P (AjB1;C)P (B1jC) + ··· + P (AjBn;C)P (BnjC) P (AjB) = P (A) P (AjC) = P (A \ B1jC) + P (A \ B2jC) + ··· + P (A \ BnjC) P (BjA) = P (B) Last Updated September 4, 2015 Special case of LOTP with B and Bc as partition: Conditional Independence A and B are conditionally independent P (A) = P (AjB)P (B) + P (AjBc)P (Bc) given C if P (A \ BjC) = P (AjC)P (BjC). -
Understanding Relative Risk, Odds Ratio, and Related Terms: As Simple As It Can Get Chittaranjan Andrade, MD
Understanding Relative Risk, Odds Ratio, and Related Terms: As Simple as It Can Get Chittaranjan Andrade, MD Each month in his online Introduction column, Dr Andrade Many research papers present findings as odds ratios (ORs) and considers theoretical and relative risks (RRs) as measures of effect size for categorical outcomes. practical ideas in clinical Whereas these and related terms have been well explained in many psychopharmacology articles,1–5 this article presents a version, with examples, that is meant with a view to update the knowledge and skills to be both simple and practical. Readers may note that the explanations of medical practitioners and examples provided apply mostly to randomized controlled trials who treat patients with (RCTs), cohort studies, and case-control studies. Nevertheless, similar psychiatric conditions. principles operate when these concepts are applied in epidemiologic Department of Psychopharmacology, National Institute research. Whereas the terms may be applied slightly differently in of Mental Health and Neurosciences, Bangalore, India different explanatory texts, the general principles are the same. ([email protected]). ABSTRACT Clinical Situation Risk, and related measures of effect size (for Consider a hypothetical RCT in which 76 depressed patients were categorical outcomes) such as relative risks and randomly assigned to receive either venlafaxine (n = 40) or placebo odds ratios, are frequently presented in research (n = 36) for 8 weeks. During the trial, new-onset sexual dysfunction articles. Not all readers know how these statistics was identified in 8 patients treated with venlafaxine and in 3 patients are derived and interpreted, nor are all readers treated with placebo. These results are presented in Table 1. -
39 Section J Basic Probability Concepts Before We Can Begin To
Section J Basic Probability Concepts Before we can begin to discuss inferential statistics, we need to discuss probability. Recall, inferential statistics deals with analyzing a sample from the population to draw conclusions about the population, therefore since the data came from a sample we can never be 100% certain the conclusion is correct. Therefore, probability is an integral part of inferential statistics and needs to be studied before starting the discussion on inferential statistics. The theoretical probability of an event is the proportion of times the event occurs in the long run, as a probability experiment is repeated over and over again. Law of Large Numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability. A sample space contains all possible outcomes of a probability experiment. EX: An event is an outcome or a collection of outcomes from a sample space. A probability model for a probability experiment consists of a sample space, along with a probability for each event. Note: If A denotes an event then the probability of the event A is denoted P(A). Probability models with equally likely outcomes If a sample space has n equally likely outcomes, and an event A has k outcomes, then Number of outcomes in A k P(A) = = Number of outcomes in the sample space n The probability of an event is always between 0 and 1, inclusive. 39 Important probability characteristics: 1) For any event A, 0 ≤ P(A) ≤ 1 2) If A cannot occur, then P(A) = 0. -
Probability and Counting Rules
blu03683_ch04.qxd 09/12/2005 12:45 PM Page 171 C HAPTER 44 Probability and Counting Rules Objectives Outline After completing this chapter, you should be able to 4–1 Introduction 1 Determine sample spaces and find the probability of an event, using classical 4–2 Sample Spaces and Probability probability or empirical probability. 4–3 The Addition Rules for Probability 2 Find the probability of compound events, using the addition rules. 4–4 The Multiplication Rules and Conditional 3 Find the probability of compound events, Probability using the multiplication rules. 4–5 Counting Rules 4 Find the conditional probability of an event. 5 Find the total number of outcomes in a 4–6 Probability and Counting Rules sequence of events, using the fundamental counting rule. 4–7 Summary 6 Find the number of ways that r objects can be selected from n objects, using the permutation rule. 7 Find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule. 8 Find the probability of an event, using the counting rules. 4–1 blu03683_ch04.qxd 09/12/2005 12:45 PM Page 172 172 Chapter 4 Probability and Counting Rules Statistics Would You Bet Your Life? Today Humans not only bet money when they gamble, but also bet their lives by engaging in unhealthy activities such as smoking, drinking, using drugs, and exceeding the speed limit when driving. Many people don’t care about the risks involved in these activities since they do not understand the concepts of probability. -
Is the Cosmos Random?
IS THE RANDOM? COSMOS QUANTUM PHYSICS Einstein’s assertion that God does not play dice with the universe has been misinterpreted By George Musser Few of Albert Einstein’s sayings have been as widely quot- ed as his remark that God does not play dice with the universe. People have naturally taken his quip as proof that he was dogmatically opposed to quantum mechanics, which views randomness as a built-in feature of the physical world. When a radioactive nucleus decays, it does so sponta- neously; no rule will tell you when or why. When a particle of light strikes a half-silvered mirror, it either reflects off it or passes through; the out- come is open until the moment it occurs. You do not need to visit a labora- tory to see these processes: lots of Web sites display streams of random digits generated by Geiger counters or quantum optics. Being unpredict- able even in principle, such numbers are ideal for cryptography, statistics and online poker. Einstein, so the standard tale goes, refused to accept that some things are indeterministic—they just happen, and there is not a darned thing anyone can do to figure out why. Almost alone among his peers, he clung to the clockwork universe of classical physics, ticking mechanistically, each moment dictating the next. The dice-playing line became emblemat- ic of the B side of his life: the tragedy of a revolutionary turned reaction- ary who upended physics with relativity theory but was, as Niels Bohr put it, “out to lunch” on quantum theory. -
Topic 1: Basic Probability Definition of Sets
Topic 1: Basic probability ² Review of sets ² Sample space and probability measure ² Probability axioms ² Basic probability laws ² Conditional probability ² Bayes' rules ² Independence ² Counting ES150 { Harvard SEAS 1 De¯nition of Sets ² A set S is a collection of objects, which are the elements of the set. { The number of elements in a set S can be ¯nite S = fx1; x2; : : : ; xng or in¯nite but countable S = fx1; x2; : : :g or uncountably in¯nite. { S can also contain elements with a certain property S = fx j x satis¯es P g ² S is a subset of T if every element of S also belongs to T S ½ T or T S If S ½ T and T ½ S then S = T . ² The universal set is the set of all objects within a context. We then consider all sets S ½ . ES150 { Harvard SEAS 2 Set Operations and Properties ² Set operations { Complement Ac: set of all elements not in A { Union A \ B: set of all elements in A or B or both { Intersection A [ B: set of all elements common in both A and B { Di®erence A ¡ B: set containing all elements in A but not in B. ² Properties of set operations { Commutative: A \ B = B \ A and A [ B = B [ A. (But A ¡ B 6= B ¡ A). { Associative: (A \ B) \ C = A \ (B \ C) = A \ B \ C. (also for [) { Distributive: A \ (B [ C) = (A \ B) [ (A \ C) A [ (B \ C) = (A [ B) \ (A [ C) { DeMorgan's laws: (A \ B)c = Ac [ Bc (A [ B)c = Ac \ Bc ES150 { Harvard SEAS 3 Elements of probability theory A probabilistic model includes ² The sample space of an experiment { set of all possible outcomes { ¯nite or in¯nite { discrete or continuous { possibly multi-dimensional ² An event A is a set of outcomes { a subset of the sample space, A ½ . -
Probability Theory Review 1 Basic Notions: Sample Space, Events
Fall 2018 Probability Theory Review Aleksandar Nikolov 1 Basic Notions: Sample Space, Events 1 A probability space (Ω; P) consists of a finite or countable set Ω called the sample space, and the P probability function P :Ω ! R such that for all ! 2 Ω, P(!) ≥ 0 and !2Ω P(!) = 1. We call an element ! 2 Ω a sample point, or outcome, or simple event. You should think of a sample space as modeling some random \experiment": Ω contains all possible outcomes of the experiment, and P(!) gives the probability that we are going to get outcome !. Note that we never speak of probabilities except in relation to a sample space. At this point we give a few examples: 1. Consider a random experiment in which we toss a single fair coin. The two possible outcomes are that the coin comes up heads (H) or tails (T), and each of these outcomes is equally likely. 1 Then the probability space is (Ω; P), where Ω = fH; T g and P(H) = P(T ) = 2 . 2. Consider a random experiment in which we toss a single coin, but the coin lands heads with 2 probability 3 . Then, once again the sample space is Ω = fH; T g but the probability function 2 1 is different: P(H) = 3 , P(T ) = 3 . 3. Consider a random experiment in which we toss a fair coin three times, and each toss is independent of the others. The coin can come up heads all three times, or come up heads twice and then tails, etc.