MAS2317 Presentation Question 1

Slides:

Advertisements

Similar presentations

Bayes Theorem. Motivating Example: Drug Tests A drug test gives a false positive 2% of the time (that is, 2% of those who test positive actually are not.

Advertisements

Let X 1, X 2,..., X n be a set of independent random variables having a common distribution, and let E[ X i ] = . then, with probability 1 Strong law.

Rare Events, Probability and Sample Size. Rare Events An event E is rare if its probability is very small, that is, if Pr{E} ≈ 0. Rare events require.

A Survey of Probability Concepts

A Survey of Probability Concepts

Foundations of Artificial Intelligence 1 Bayes’ Rule - Example  Medical Diagnosis  suppose we know from statistical data that flu causes fever in 80%

Statistics Lecture 6. Last day: Probability rules Today: Conditional probability Suggested problems: Chapter 2: 45, 47, 59, 63, 65.

Partitions Law of Total Probability Bayes’ Rule

4. Convergence of random variables  Convergence in probability  Convergence in distribution  Convergence in quadratic mean  Properties  The law of.

Chapter 4: Probability (Cont.) In this handout: Total probability rule Bayes’ rule Random sampling from finite population Rule of combinations.

Today Today: Some more counting examples; Start Chapter 2 Important Sections from Chapter 1: ; Please read Reading: –Assignment #2 is up.

PROBABILITY AND SAMPLES: THE DISTRIBUTION OF SAMPLE MEANS.

The moment generating function of random variable X is given by Moment generating function.

2003/04/24 Chapter 5 1頁1頁 Chapter 5 : Sums of Random Variables & Long-Term Averages 5.1 Sums of Random Variables.

Baye’s Rule and Medical Screening Tests. Baye’s Rule Baye’s Rule is used in medicine and epidemiology to calculate the probability that an individual.

1 The probability that a medical test will correctly detect the presence of a certain disease is 98%. The probability that this test will correctly detect.

1. Probability of Equally Likely Outcomes 2. Complement Rule 1.

INFORMATION THEORY BAYESIAN STATISTICS II Thomas Tiahrt, MA, PhD CSC492 – Advanced Text Analytics.

Sample Distribution Models for Means and Proportions

Distributions of Sample Means and Sample Proportions BUSA 2100, Sections 7.0, 7.1, 7.4, 7.5, 7.6.

Conditional Probability Brian Carrico Nov 5, 2009.

Quiz 2 - Review. Descriptive Statistics Be able to interpret: -Box Plots and Histograms -Mean, Median, Standard Deviation, and Percentiles.

COUNTING POPULATIONS Estimations & Random Samples.

Ch Counting Techniques Product Rule If the first element or object of an ordered pair can be used in n 1 ways, and for each of these n1 ways.

Bayes’ Theorem Bayes’ Theorem allows us to calculate the conditional probability one way (e.g., P(B|A) when we know the conditional probability the other.

 P(A c ∩ B) + P(A ∩ B)=P(B)  P(A c ∩ B) and P(A ∩ B) are called joint probability.  P(A) and P(B) are called marginal probability.  P(A|B) and P(B|A)

Baye’s Theorem Working with Conditional Probabilities.

Bayes’ Theorem -- Partitions Given two events, R and S, if P(R  S) =1 P(R  S) =0 then we say that R and S partition the sample space. More than 2 events.

6.3 Bayes Theorem. We can use Bayes Theorem… …when we know some conditional probabilities, but wish to know others. For example: We know P(test positive|have.

NLP. Introduction to NLP Formula for joint probability –p(A,B) = p(B|A)p(A) –p(A,B) = p(A|B)p(B) Therefore –p(B|A)=p(A|B)p(B)/p(A) Bayes’ theorem is.

Central Limit Theorem-CLT MM4D1. Using simulation, students will develop the idea of the central limit theorem.

Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-1 Developing a Sampling Distribution Assume there is a population … Population size N=4.

Population and Sample The entire group of individuals that we want information about is called population. A sample is a part of the population that we.

Areej Jouhar & Hafsa El-Zain Biostatistics BIOS 101 Foundation year.

© 2013 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems Introductory Statistics: Exploring the World through.

Estimation Chapter 8. Estimating µ When σ Is Known.

Confidence Interval & Unbiased Estimator Review and Foreword.

IE 300, Fall 2012 Richard Sowers IESE. 8/30/2012 Goals: Rules of Probability Counting Equally likely Some examples.

1 Sampling distributions The probability distribution of a statistic is called a sampling distribution. : the sampling distribution of the mean.

Week 21 Rules of Probability for all Corollary: The probability of the union of any two events A and B is Proof: … If then, Proof:

Distributions of Sample Means. z-scores for Samples  What do I mean by a “z-score” for a sample? This score would describe how a specific sample is.

Central Limit Theorem Let X 1, X 2, …, X n be n independent, identically distributed random variables with mean  and standard deviation . For large n:

Probability Introduction. Producing DataSlide #2 A Series of Examples 1) Assume I have a box with 12 red balls and 8 green balls. If I mix the balls in.

STAT 240 PROBLEM SOLVING SESSION #2. Conditional Probability.

Sampling Distributions Chapter 9 Central Limit Theorem.

Looking at Bayes’ Theorem with a Tree Diagram An alternate way of “reversing the conditioning” on conditional probabilities.

Conditional Probability

Looking at Bayes’ Theorem with a Tree Diagram

Chapter 5: Bayes’ Theorem (And Additional Applications)

Basic Statistics and the Law of Large Numbers

Chapter 2 Appendix Basic Statistics and the Law of Large Numbers.

Hypothesis Testing.

Psychology 202a Advanced Psychological Statistics

Bayesian Notions and False Positives

In-Class Exercise: Bayes’ Theorem

Mathematical representation of Bayes theorem.

Basic Probability Concepts

Executive Director and Endowed Chair

POINT ESTIMATOR OF PARAMETERS

Executive Director and Endowed Chair

Baye’s Rule and Medical Screening Tests

Probability and Statistics

CHAPTER 15 SUMMARY Chapter Specifics

COnDITIONAL Probability

Random Samples Random digit table

Chance Errors in Sampling (Dr. Monticino)

Probability and Statistics

Power Problems.

Discrete Math for CS CMPSC 360 LECTURE 34 Last time:

Presentation transcript:

MAS2317 Presentation Question 1 Ryan Sheridan

Question Suppose that 1% of the population have a disease D. A diagnostic test S, designed to detect the disease, has the following accuracy: Pr(S positive|D) = 0.95 and Pr (S positive|Dc) = 0.05. If 100 people were tested at random, how many would we expect to test positive and of those that do, about how many would we expect to have the disease?

How many would we expect to be positive? Using the Law of Total Probability, Pr 𝐹 = 𝑖=1 𝑛 Pr 𝐹 𝐸 𝑖 Pr⁡( 𝐸 𝑖 ) , we get; Pr +𝑣𝑒 = Pr +𝑣𝑒 𝐷 Pr 𝐷 + Pr +𝑣𝑒 𝐷 𝑐 Pr⁡( 𝐷 𝑐 ) =0.95×0.01+0.05×0.99 =0.059 So out of 100 random people being tested we would expect about 6 people test positive regardless of if they have the disease or not.

Now to find how many of those 6 people we expect to actually have the disease. Using Bayes’ theorem we get, Pr 𝐷 +𝑣𝑒 = Pr +𝑣𝑒 𝐷 Pr⁡(𝐷) Pr +𝑣𝑒 𝐷 Pr 𝐷 +𝑃𝑟 +𝑣𝑒 𝐷 𝑐 Pr⁡( 𝐷 𝑐 ) = 0.95×0.01 0.95×0.01+0.05×0.99 =0.161 So we now have the probability of getting a positive test and the probability of having the disease given a positive test, now to find the number of people to actually have the disease: 0.059×0.161×100%=0.9499% So we would expect only one person to actually have the disease given the test was positive for them.