PHP 2510 Probability Part 3. Calculate probability by counting (cont ) Conditional probability. Law of total probability. Bayes Theorem.
|
|
- Suzanna Perkins
- 5 years ago
- Views:
Transcription
1 PHP 2510 Probability Part 3 Calculate probability by counting (cont ) Conditional probability Law of total probability Bayes Theorem Independence PHP 2510 Sept 22,
2 Last time... Count the number of ways to randomly sample r subjects (e.g. r labeled balls) Context: From a finite population (from an urn containing n balls) without replacement. If ordering is relevant (permutations), n (n 1)... (n r + 1) = n! (n r)!. If ordering does not matter (combinations), ( ) n! n (n r)!r! =. r PHP 2510 Sept 22,
3 Capture-recapture methods: Example 1 Suppose there are 100 bobcats in a certain state park. A scientist captures 10 of them, tags them, and releases them back to their habitat. A week later, he recaptures 20. What is the probability that 4 of the recaptured bobcats 20 have tags? Denominator: how many ways to draw the 20 bobcats from the population of 100? Numerator: In the recaptured sample of 20, how many ways to 1. Draw 4 tagged bobcats from the 10 that have tags 2. Draw 16 non-tagged bobcats from the 90 that do not have tags Ans: ( 10 4 )( ( ) ).08 PHP 2510 Sept 22,
4 Capture-recapture methods: Example 2 The most common use of capture-recapture experiments is to estimate a population size. In the previous example, assume the population size is unknown; denote it by N. The biologist captures 10 bobcats and tags them. A week later he captures 20 and finds that 4 are tagged. To find the population size, we can find the value of N that, in some sense, is the most consistent with the observed data. Hence we find N that maximizes the expression ( 10 )( N 10 ) ( N 20) In this case it turns out that N = 50 maximizes the probability. PHP 2510 Sept 22,
5 Counting possible allocations into several categories Property. The number of ways of grouping n objects into r classes is ( ) n! n 1! n 2! n r! = n n 1, n 2,..., n r This number is called the multinomial coefficient PHP 2510 Sept 22,
6 Example: Allocating committee members A committee of 7 members is divided into three subcommittees of size 4, 2 and 1. How many distinct ways can the subcommittees be formed? Numerator: number of ways of arranging the 7 members Denominator: number of ways to allocate the subcommittees Ans: ( ) 7 4, 2, 1 = 7! 4!2!1! = 105 PHP 2510 Sept 22,
7 Example: Genomic sequences A DNA molecule is a sequence of nucleotides labeled A, C, G, and T. The entire molecule can be millions of units long. Sometimes researchers are interested in the arrangement of nucleotides at a specific location on the DNA molecule. How many ways can the nucleotides {A, A, G, G, G, T, T, T, T } be arranged in a sequence that is 9 units long? Ans: ( ) 9 2, 3, 4 = 9! 2!3!4! = 1260 Why is this a problem of combinations, not permutations? PHP 2510 Sept 22,
8 Conditional probability Consider screening randomly-selected individuals for HIV. For each person we can define two events: the test status and the HIV status. H +, H = HIV status T +, T = test status PHP 2510 Sept 22,
9 In practice, we frequently will be interested in specific conditional probabilities: P (H + T + ) = prob of being HIV+, conditional on having a positive test P (T + H + ) = prob of having a positive test, conditional on being HIV+ PHP 2510 Sept 22,
10 Definition of conditional probability For any two events A and B, where P (B) > 0, the conditional probability of event A, given that event B already has occurred, is given by P (A B) = P (A B). P (B) The idea is that if we are given that event B occurred, the relevant sample space becomes B rather than the original sample size Ω. PHP 2510 Sept 22,
11 Example 1: HIV testing Recall the HIV testing situation. The sample space is Ω = {(H +, T + ), (H +, T ), (H, T + ), (H, T )} Suppose the probabilities are as follows: H + H T T Find the following: P (H + ), P (T + ), P (H + T + ), P (H + T + ) PHP 2510 Sept 22,
12 Computing conditional probabilities What is the probability of being HIV+, given that the test result is positive? (This is called the predictive value of a positive test). P (H + T + ) = =.71 What is the probability of being HIV, given that the test result is positive? Use the complement law. Ans:.29 What is the probability of testing positive, given that true HIV status is positive? (This is called sensitivity of a diagnostic test). P (T + H + ) = =.91 PHP 2510 Sept 22,
13 Multiplication law A direct result of this definition is the multiplication law, which can be used to find a joint probability of A B: P (A B) = P (B) P (A B) The multiplication law can be used to calculate a joint probability P (A B) when P (A B) and P (B) known. Example. An urn contains 5 red balls and 6 white balls. You select two balls at random, without replacement. What is the probability of selecting 2 red balls? Ans: =.18 PHP 2510 Sept 22,
14 Example. Suppose that if it is cloudy (B), the probability that it is raining (A) is 0.3, and that the probability that it is cloudy is P (B) is 0.2. What is the probability that it is cloudy and raining? Ans: Example. In a particular community, the prevalence of HIV is.10. Among those with HIV, it is found that 40% also have hepatitis C (HCV). What is the probability that a randomly selected person is infected with both HIV and HCV? Ans: PHP 2510 Sept 22,
15 Law of total probability Suppose the events B 1, B 2,..., B n are disjoint and exhaustive. By disjoint, we mean that B i B j = for any i j. By exhaustive, we mean that Then for any event A, P (B 1 B 2 B n ) = 1 P (A) = P (A B 1 ) + P (A B 2 ) + + P (A B n ) = n P (A B i ) i=1 Because P (A B i ) = P (A B i )P (B i ), we also have P (A) = P (A B 1 )P (B 1 ) + P (A B 2 )P (B 2 ) + + P (A B n )P (B n ) = n P (A B i )P (B i ) i=1 PHP 2510 Sept 22,
16 Law of total probability, in words Says that the probability of an event A is a weighted average of conditional probabilities taken over a set of mutually exclusive and exhaustive events. The conditional probabilities are P (A B i ) and they are weighted by P (B i ). PHP 2510 Sept 22,
17 Application of the law of total probability Consider MRI for breast CA, where M = 1, 2, 3 is true degree of malignancy (0=none, 1=benign, 2=malignant). Suppose prevalence of each underlying status, for the population being screened, is 0.80 (none), 0.15 (benign), 0.05 (malignant). Upon radiology scan, a biopsy is either ordered (B) or not ordered (B c ), depending on the judgment of the radiologist. Suppose a particular radiologist orders biopsies with the following probabilities: P (B M = 0) =.10 P (B M = 1) =.40 P (B M = 2) =.90 What is the probability that a patient undergoing breast MRI will be referred for a biopsy? Ans =.19 PHP 2510 Sept 22,
18 Bayes Theorem Let B 1,..., B n be a set of disjoint and exhaustive events. Then for some event A, the conditional probability P (B j A) is P (B j A) = P (B j A) P (A) = P (A B j )P (B j ) n i=1 P (A B i)p (B i ) Example. In the previous example, among those referred for biopsy, what is the probability of having a benign tumor? Ans:.42. What is the probability of having a malignant tumor? Ans:.24. PHP 2510 Sept 22,
19 Independent events Two events A and B are said to be independent if P (A B) = P (A)P (B) The intuition behind this can be seen as follows. We can say that A and B are independent if conditioning on B does not affect the probability of A occurring. PHP 2510 Sept 22,
20 In other word, P (A B) = P (A) and P (B A) = P (B). But we can rewrite P (A B) as P (A B) P (B), which implies the definition above. Note that A B = dose NOT imply that A and B are independent. PHP 2510 Sept 22,
21 Calculating probabilities for independent events Example 1. Suppose the population prevalence of TB (pulmonary tuberculosis) is.01. Two randomly-selected individuals are tested for TB. What is the probability that: (a) both have TB? (b) neither has TB? (c) exactly one has TB? Ans: (a).001; (b).9801; (c).0198 Example 2. In randomly selected cohabiting couples, it is found that the proportion of couples where both members have TB is.01. Is the presence of TB in one member of a couple independent of presence of TB in the other member? PHP 2510 Sept 22,
22 Example 3. A gene expression array has 10,000 cells that each measure RNA expression of individual genes. The false positive rate for any given cell i.e., the probability of indicating gene activity when it is in fact absent is 1 in The expression array is applied to a neutral medium, where it is known that none of the genes on the array will be active. What is the probability that the array will correctly indicate no activity for each of the 10,000 genes? Ans:.135 Example 4. Suppose that the probability of contracting HIV in one act of sexual intercourse is 1 in 500. If a person has 100 sexual encounters with an HIV-infected individual, what is the probability of contracting HIV? Ans:.181 PHP 2510 Sept 22,
MATH 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 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 information1 Preliminaries Sample Space and Events Interpretation of Probability... 13
Summer 2017 UAkron Dept. of Stats [3470 : 461/561] Applied Statistics Ch 2: Probability Contents 1 Preliminaries 3 1.1 Sample Space and Events...........................................................
More informationLecture 01: Introduction
Lecture 01: Introduction Dipankar Bandyopadhyay, Ph.D. BMTRY 711: Analysis of Categorical Data Spring 2011 Division of Biostatistics and Epidemiology Medical University of South Carolina Lecture 01: Introduction
More information1 Probability Theory. 1.1 Introduction
1 Probability Theory Probability theory is used as a tool in statistics. It helps to evaluate the reliability of our conclusions about the population when we have only information about a sample. Probability
More informationJoint, Conditional, & Marginal Probabilities
Joint, Conditional, & Marginal Probabilities The three axioms for probability don t discuss how to create probabilities for combined events such as P [A B] or for the likelihood of an event A given that
More informationAxioms of Probability
MAT 2379 3X (Spring 2012) (Introduction to Probability (part II)) The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and
More informationRelative Risks (RR) and Odds Ratios (OR) 20
BSTT523: Pagano & Gavreau, Chapter 6 1 Chapter 6: Probability slide: Definitions (6.1 in P&G) 2 Experiments; trials; probabilities Event operations 4 Intersection; Union; Complement Venn diagrams Conditional
More informationElements of probability theory
The role of probability theory in statistics We collect data so as to provide evidentiary support for answers we give to our many questions about the world (and in our particular case, about the business
More informationLecture 3: Probability
Lecture 3: Probability 28th of October 2015 Lecture 3: Probability 28th of October 2015 1 / 36 Summary of previous lecture Define chance experiment, sample space and event Introduce the concept of the
More informationMultiple Choice Practice Set 1
Multiple Choice Practice Set 1 This set of questions covers material from Chapter 1. Multiple choice is the same format as for the midterm. Q1. Two events each have probability 0.2 of occurring and are
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 informationBasic Statistics and Probability Chapter 3: Probability
Basic Statistics and Probability Chapter 3: Probability Events, Sample Spaces and Probability Unions and Intersections Complementary Events Additive Rule. Mutually Exclusive Events Conditional Probability
More informationMAT2377. Ali Karimnezhad. Version September 9, Ali Karimnezhad
MAT2377 Ali Karimnezhad Version September 9, 2015 Ali Karimnezhad Comments These slides cover material from Chapter 1. In class, I may use a blackboard. I recommend reading these slides before you come
More information2.6 Tools for Counting sample points
2.6 Tools for Counting sample points When the number of simple events in S is too large, manual enumeration of every sample point in S is tedious or even impossible. (Example) If S contains N equiprobable
More informationNaive Bayes classification
Naive Bayes classification Christos Dimitrakakis December 4, 2015 1 Introduction One of the most important methods in machine learning and statistics is that of Bayesian inference. This is the most fundamental
More informationToday we ll discuss ways to learn how to think about events that are influenced by chance.
Overview Today we ll discuss ways to learn how to think about events that are influenced by chance. Basic probability: cards, coins and dice Definitions and rules: mutually exclusive events and independent
More informationReview. More Review. Things to know about Probability: Let Ω be the sample space for a probability measure P.
1 2 Review Data for assessing the sensitivity and specificity of a test are usually of the form disease category test result diseased (+) nondiseased ( ) + A B C D Sensitivity: is the proportion of diseased
More informationProbability. Introduction to Biostatistics
Introduction to Biostatistics Probability Second Semester 2014/2015 Text Book: Basic Concepts and Methodology for the Health Sciences By Wayne W. Daniel, 10 th edition Dr. Sireen Alkhaldi, BDS, MPH, DrPH
More informationMath 140 Introductory Statistics
5. Models of Random Behavior Math 40 Introductory Statistics Professor Silvia Fernández Chapter 5 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Outcome: Result or answer
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Professor Silvia Fernández Lecture 8 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. 5.1 Models of Random Behavior Outcome: Result or answer
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 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 informationBayes Theorem & Diagnostic Tests Screening Tests
Bayes heore & Diagnostic ests Screening ests Box contains 2 red balls and blue ball Box 2 contains red ball and 3 blue balls A coin is tossed. If Head turns up a ball is drawn fro Box, and if ail turns
More informationProbability II. Patrick Breheny. February 16. Basic rules (cont d) Advanced rules Summary
Probability II Patrick Breheny February 16 Patrick Breheny University of Iowa Introduction to Biostatistics (BIOS 4120) 1 / 48 Balls in urns Basic rules (cont d) The multiplication rule Imagine a random
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 informationConditional Probability and Independence
Conditional Probability and Independence September 3, 2009 1 Restricting the Sample Space - Conditional Probability How do we modify the probability of an event in light of the fact that something is known?
More informationChance, too, which seems to rush along with slack reins, is bridled and governed by law (Boethius, ).
Chapter 2 Probability Chance, too, which seems to rush along with slack reins, is bridled and governed by law (Boethius, 480-524). Blaise Pascal (1623-1662) Pierre de Fermat (1601-1665) Abraham de Moivre
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 informationLecture Stat 302 Introduction to Probability - Slides 5
Lecture Stat 302 Introduction to Probability - Slides 5 AD Jan. 2010 AD () Jan. 2010 1 / 20 Conditional Probabilities Conditional Probability. Consider an experiment with sample space S. Let E and F be
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 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 informationTest One Mathematics Fall 2009
Test One Mathematics 35.2 Fall 29 TO GET FULL CREDIT YOU MUST SHOW ALL WORK! I have neither given nor received aid in the completion of this test. Signature: pts. 2 pts. 3 5 pts. 2 pts. 5 pts. 6(i) pts.
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 informationLecture 2. Conditional Probability
Math 408 - Mathematical Statistics Lecture 2. Conditional Probability January 18, 2013 Konstantin Zuev (USC) Math 408, Lecture 2 January 18, 2013 1 / 9 Agenda Motivation and Definition Properties of Conditional
More informationCHAPTER 4. Probability is used in inference statistics as a tool to make statement for population from sample information.
CHAPTER 4 PROBABILITY Probability is used in inference statistics as a tool to make statement for population from sample information. Experiment is a process for generating observations Sample space is
More informationInstitution. 3. Which sextants, contain post-biopsy hemorrhage (irregular areas of increased signal on T1 images): (Check all that apply)
M ACRIN 9 MRI/MRSI Imaging Technical Form Institution ACRIN Study 9 PLACE LABEL HERE Institution No Participant Initials Case No Instructions: The study designated Radiologist is to complete this form
More informationProbability and Probability Distributions. Dr. Mohammed Alahmed
Probability and Probability Distributions 1 Probability and Probability Distributions Usually we want to do more with data than just describing them! We might want to test certain specific inferences about
More informationECE353: Probability and Random Processes. Lecture 3 - Independence and Sequential Experiments
ECE353: Probability and Random Processes Lecture 3 - Independence and Sequential Experiments Xiao Fu School of Electrical Engineering and Computer Science Oregon State University E-mail: xiao.fu@oregonstate.edu
More informationProbability and Statistics. Terms and concepts
Probability and Statistics Joyeeta Dutta Moscato June 30, 2014 Terms and concepts Sample vs population Central tendency: Mean, median, mode Variance, standard deviation Normal distribution Cumulative distribution
More informationConditional Probability (cont...) 10/06/2005
Conditional Probability (cont...) 10/06/2005 Independent Events Two events E and F are independent if both E and F have positive probability and if P (E F ) = P (E), and P (F E) = P (F ). 1 Theorem. If
More informationWhat is the probability of getting a heads when flipping a coin
Chapter 2 Probability Probability theory is a branch of mathematics dealing with chance phenomena. The origins of the subject date back to the Italian mathematician Cardano about 1550, and French mathematicians
More informationMATH2206 Prob Stat/20.Jan Weekly Review 1-2
MATH2206 Prob Stat/20.Jan.2017 Weekly Review 1-2 This week I explained the idea behind the formula of the well-known statistic standard deviation so that it is clear now why it is a measure of dispersion
More informationELEG 3143 Probability & Stochastic Process Ch. 1 Probability
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Applications Elementary Set Theory Random
More informationSTAC51: Categorical data Analysis
STAC51: Categorical data Analysis Mahinda Samarakoon January 26, 2016 Mahinda Samarakoon STAC51: Categorical data Analysis 1 / 32 Table of contents Contingency Tables 1 Contingency Tables Mahinda Samarakoon
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 informationConditional Probability
Chapter 3 Conditional Probability 3.1 Definition of conditional probability In spite of our misgivings, let us persist with the frequency definition of probability. Consider an experiment conducted N times
More informationIntroduction to Probability Theory, Algebra, and Set Theory
Summer School on Mathematical Philosophy for Female Students Introduction to Probability Theory, Algebra, and Set Theory Catrin Campbell-Moore and Sebastian Lutz July 28, 2014 Question 1. Draw Venn diagrams
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Problem Set 3: Solutions Fall 2007
UC Berkeley Department of Electrical Engineering and Computer Science EE 126: Probablity and Random Processes Problem Set 3: Solutions Fall 2007 Issued: Thursday, September 13, 2007 Due: Friday, September
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 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 informationPart 1: Logic and Probability
Part 1: Logic and Probability In some sense, probability subsumes logic: While a probability can be seen as a measure of degree of truth a real number between 0 and 1 logic deals merely with the two extreme
More informationConditional Probability Solutions STAT-UB.0103 Statistics for Business Control and Regression Models
Conditional Probability Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Counting (Review) 1. There are 10 people in a club. How many ways are there to choose the following:
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 informationCS37300 Class Notes. Jennifer Neville, Sebastian Moreno, Bruno Ribeiro
CS37300 Class Notes Jennifer Neville, Sebastian Moreno, Bruno Ribeiro 2 Background on Probability and Statistics These are basic definitions, concepts, and equations that should have been covered in your
More information2 Chapter 2: Conditional Probability
STAT 421 Lecture Notes 18 2 Chapter 2: Conditional Probability Consider a sample space S and two events A and B. For example, suppose that the equally likely sample space is S = {0, 1, 2,..., 99} and A
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 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 information(3) Review of Probability. ST440/540: Applied Bayesian Statistics
Review of probability The crux of Bayesian statistics is to compute the posterior distribution, i.e., the uncertainty distribution of the parameters (θ) after observing the data (Y) This is the conditional
More informationNotes Week 2 Chapter 3 Probability WEEK 2 page 1
Notes Week 2 Chapter 3 Probability WEEK 2 page 1 The sample space of an experiment, sometimes denoted S or in probability theory, is the set that consists of all possible elementary outcomes of that experiment
More informationProbability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary)
Chapter 14 From Randomness to Probability How to measure a likelihood of an event? How likely is it to answer correctly one out of two true-false questions on a quiz? Is it more, less, or equally likely
More informationProbability: Why do we care? Lecture 2: Probability and Distributions. Classical Definition. What is Probability?
Probability: Why do we care? Lecture 2: Probability and Distributions Sandy Eckel seckel@jhsph.edu 22 April 2008 Probability helps us by: Allowing us to translate scientific questions into mathematical
More informationSome Concepts of Probability (Review) Volker Tresp Summer 2018
Some Concepts of Probability (Review) Volker Tresp Summer 2018 1 Definition There are different way to define what a probability stands for Mathematically, the most rigorous definition is based on Kolmogorov
More informationLectures for STP 421: Probability Theory. Jay Taylor
Lectures for STP 421: Probability Theory Jay Taylor February 27, 2012 Contents 1 Overview and Conceptual Foundations of Probability 5 1.1 Deterministic and Statistical Regularity.......................
More informationConditional Probability, Independence and Bayes Theorem Class 3, Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence of events. 2.
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 informationSolution to HW 12. Since B and B 2 form a partition, we have P (A) = P (A B 1 )P (B 1 ) + P (A B 2 )P (B 2 ). Using P (A) = 21.
Solution to HW 12 (1) (10 pts) Sec 12.3 Problem A screening test for a disease shows a positive result in 92% of all cases when the disease is actually present and in 7% of all cases when it is not. Assume
More informationJoint, Conditional, & Marginal Probabilities
Joint, Conditional, & Marginal Probabilities Statistics 110 Summer 2006 Copyright c 2006 by Mark E. Irwin Joint, Conditional, & Marginal Probabilities The three axioms for probability don t discuss how
More informationProbability. Patrick Breheny. September 1. Events and probability Working with probabilities Additional theorems/rules Summary
Probability Patrick Breheny September 1 Patrick Breheny University of Iowa Biostatistical Methods I (BIOS 5710) 1 / 52 Probability Events and probability Mathematical definitions Meaning and interpretation
More informationPROBABILITY.
PROBABILITY PROBABILITY(Basic Terminology) Random Experiment: If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes,
More informationDiscrete Probability
Discrete Probability Mark Muldoon School of Mathematics, University of Manchester M05: Mathematical Methods, January 30, 2007 Discrete Probability - p. 1/38 Overview Mutually exclusive Independent More
More informationLecture 2: Probability. Readings: Sections Statistical Inference: drawing conclusions about the population based on a sample
Lecture 2: Probability Readings: Sections 5.1-5.3 1 Introduction Statistical Inference: drawing conclusions about the population based on a sample Parameter: a number that describes the population a fixed
More informationModule 2 : Conditional Probability
Module 2 : Conditional Probability Ruben Zamar Department of Statistics UBC January 16, 2017 Ruben Zamar Department of Statistics UBC Module () 2 January 16, 2017 1 / 61 MOTIVATION The outcome could be
More information2.4 Conditional Probability
2.4 Conditional Probability The probabilities assigned to various events depend on what is known about the experimental situation when the assignment is made. Example: Suppose a pair of dice is tossed.
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 informationConditional Probability. CS231 Dianna Xu
Conditional Probability CS231 Dianna Xu 1 Boy or Girl? A couple has two children, one of them is a girl. What is the probability that the other one is also a girl? Assuming 50/50 chances of conceiving
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 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: CONDITIONING, BAYES THEOREM [DEVORE 2.4]
PROBABILITY: CONDITIONING, BAYES THEOREM [DEVORE 2.4] CONDITIONAL PROBABILITY: Let events E, F be events in the sample space Ω of an experiment. Then: The conditional probability of F given E, P(F E),
More informationn N CHAPTER 1 Atoms Thermodynamics Molecules Statistical Thermodynamics (S.T.)
CHAPTER 1 Atoms Thermodynamics Molecules Statistical Thermodynamics (S.T.) S.T. is the key to understanding driving forces. e.g., determines if a process proceeds spontaneously. Let s start with entropy
More informationCourse: ESO-209 Home Work: 1 Instructor: Debasis Kundu
Home Work: 1 1. Describe the sample space when a coin is tossed (a) once, (b) three times, (c) n times, (d) an infinite number of times. 2. A coin is tossed until for the first time the same result appear
More informationDiscovering molecular pathways from protein interaction and ge
Discovering molecular pathways from protein interaction and gene expression data 9-4-2008 Aim To have a mechanism for inferring pathways from gene expression and protein interaction data. Motivation Why
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 informationLecture 2: Probability and Distributions
Lecture 2: Probability and Distributions Ani Manichaikul amanicha@jhsph.edu 17 April 2007 1 / 65 Probability: Why do we care? Probability helps us by: Allowing us to translate scientific questions info
More informationLecture 04: Conditional Probability. Lisa Yan July 2, 2018
Lecture 04: Conditional Probability Lisa Yan July 2, 2018 Announcements Problem Set #1 due on Friday Gradescope submission portal up Use Piazza No class or OH on Wednesday July 4 th 2 Summary from last
More informationStatistics for Business and Economics
Statistics for Business and Economics Basic Probability Learning Objectives In this lecture(s), you learn: Basic probability concepts Conditional probability To use Bayes Theorem to revise probabilities
More informationSTAT509: Probability
University of South Carolina August 20, 2014 The Engineering Method and Statistical Thinking The general steps of engineering method are: 1. Develop a clear and concise description of the problem. 2. Identify
More informationLecture 8: Conditional probability I: definition, independence, the tree method, sampling, chain rule for independent events
Lecture 8: Conditional probability I: definition, independence, the tree method, sampling, chain rule for independent events Discrete Structures II (Summer 2018) Rutgers University Instructor: Abhishek
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 informationLecture 3. January 7, () Lecture 3 January 7, / 35
Lecture 3 January 7, 2013 () Lecture 3 January 7, 2013 1 / 35 Outline This week s lecture: Fast review of last week s lecture: Conditional probability. Partition, Partition theorem. Bayes theorem and its
More informationENGR 200 ENGR 200. What did we do last week?
ENGR 200 What did we do last week? Definition of probability xioms of probability Sample space robability laws Conditional probability ENGR 200 Lecture 3: genda. Conditional probability 2. Multiplication
More informationChapter 4. Probability Theory. 4.1 Probability theory Events
Chapter 4 Probability Theory Probability theory is a branch of mathematics that is an essential component of statistics. It originally evolved from efforts to understand the odds and probabilities involved
More informationCINQA Workshop Probability Math 105 Silvia Heubach Department of Mathematics, CSULA Thursday, September 6, 2012
CINQA Workshop Probability Math 105 Silvia Heubach Department of Mathematics, CSULA Thursday, September 6, 2012 Silvia Heubach/CINQA 2012 Workshop Objectives To familiarize biology faculty with one of
More information2030 LECTURES. R. Craigen. Inclusion/Exclusion and Relations
2030 LECTURES R. Craigen Inclusion/Exclusion and Relations The Principle of Inclusion-Exclusion 7 ROS enumerates the union of disjoint sets. What if sets overlap? Some 17 out of 30 students in a class
More information3. The Multivariate Hypergeometric Distribution
1 of 6 7/16/2009 6:47 AM Virtual Laboratories > 12. Finite Sampling Models > 1 2 3 4 5 6 7 8 9 3. The Multivariate Hypergeometric Distribution Basic Theory As in the basic sampling model, we start with
More informationClass 26: review for final exam 18.05, Spring 2014
Probability Class 26: review for final eam 8.05, Spring 204 Counting Sets Inclusion-eclusion principle Rule of product (multiplication rule) Permutation and combinations Basics Outcome, sample space, event
More informationUncertainty. Chapter 13
Uncertainty Chapter 13 Uncertainty Let action A t = leave for airport t minutes before flight Will A t get me there on time? Problems: 1. partial observability (road state, other drivers' plans, noisy
More informationChapter 2 PROBABILITY SAMPLE SPACE
Chapter 2 PROBABILITY Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation,
More information6.047 / Computational Biology: Genomes, Networks, Evolution Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.047 / 6.878 Computational Biology: Genomes, etworks, Evolution Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationChapter 4 Probability
4-1 Review and Preview Chapter 4 Probability 4-2 Basic Concepts of Probability 4-3 Addition Rule 4-4 Multiplication Rule: Basics 4-5 Multiplication Rule: Complements and Conditional Probability 4-6 Counting
More information