Problem A newly married couple plans to have four children and would like to have three girls and a boy. What are the chances (probability) their desire.

Slides:



Advertisements
Similar presentations
Fundamentals of Probability
Advertisements

Presentation on Probability Distribution * Binomial * Chi-square
Acknowledgement: Thanks to Professor Pagano
Lecture (7) Random Variables and Distribution Functions.
Chapter 5 Discrete Random Variables and Probability Distributions
Chapter 4 Probability and Probability Distributions
COUNTING AND PROBABILITY
ฟังก์ชั่นการแจกแจงความน่าจะเป็น แบบไม่ต่อเนื่อง Discrete Probability Distributions.
Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Statistics.
Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 5-1 Statistics for Managers Using Microsoft® Excel 5th Edition.
Chapter 4: Probabilistic features of certain data Distributions Pages
Chapter 4 Discrete Random Variables and Probability Distributions
Chapter 5 Basic Probability Distributions
Chapter 4 Probability Distributions
Probability and Probability Distributions
CHAPTER 6 Statistical Analysis of Experimental Data
Discrete Probability Distributions
Lecture Slides Elementary Statistics Twelfth Edition
Chapter 5 Discrete Random Variables and Probability Distributions
McGraw-Hill Ryerson Copyright © 2011 McGraw-Hill Ryerson Limited. Adapted by Peter Au, George Brown College.
McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 4 and 5 Probability and Discrete Random Variables.
Chap 5-1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall Chapter 5 Discrete Probability Distributions Business Statistics: A First.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. Review and Preview This chapter combines the methods of descriptive statistics presented in.
Slide 1 Copyright © 2004 Pearson Education, Inc..
Ex St 801 Statistical Methods Probability and Distributions.
Probability The definition – probability of an Event Applies only to the special case when 1.The sample space has a finite no.of outcomes, and 2.Each.
Theory of Probability Statistics for Business and Economics.
Sullivan – Fundamentals of Statistics – 2 nd Edition – Chapter 11 Section 1 – Slide 1 of 34 Chapter 11 Section 1 Random Variables.
Quality Improvement PowerPoint presentation to accompany Besterfield, Quality Improvement, 9e PowerPoint presentation to accompany Besterfield, Quality.
OPIM 5103-Lecture #3 Jose M. Cruz Assistant Professor.
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Basic Business Statistics.
Random Variables. A random variable X is a real valued function defined on the sample space, X : S  R. The set { s  S : X ( s )  [ a, b ] is an event}.
 A probability function is a function which assigns probabilities to the values of a random variable.  Individual probability values may be denoted by.
Worked examples and exercises are in the text STROUD PROGRAMME 28 PROBABILITY.
BINOMIALDISTRIBUTION AND ITS APPLICATION. Binomial Distribution  The binomial probability density function –f(x) = n C x p x q n-x for x=0,1,2,3…,n for.
Biostatistics Class 3 Discrete Probability Distributions 2/8/2000.
Binomial Experiment A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
Summary -1 Chapters 2-6 of DeCoursey. Basic Probability (Chapter 2, W.J.Decoursey, 2003) Objectives: -Define probability and its relationship to relative.
Computing Fundamentals 2 Lecture 6 Probability Lecturer: Patrick Browne
Dr. Ahmed Abdelwahab Introduction for EE420. Probability Theory Probability theory is rooted in phenomena that can be modeled by an experiment with an.
Measuring chance Probabilities FETP India. Competency to be gained from this lecture Apply probabilities to field epidemiology.
Probability Theory Modelling random phenomena. Permutations the number of ways that you can order n objects is: n! = n(n-1)(n-2)(n-3)…(3)(2)(1) Definition:
Lecture 7 Dustin Lueker.  Experiment ◦ Any activity from which an outcome, measurement, or other such result is obtained  Random (or Chance) Experiment.
Chapter 2: Probability. Section 2.1: Basic Ideas Definition: An experiment is a process that results in an outcome that cannot be predicted in advance.
STROUD Worked examples and exercises are in the text Programme 29: Probability PROGRAMME 29 PROBABILITY.
Chapter 5 Probability Distributions 5-1 Overview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance and Standard Deviation.
Lecture-6 Models for Data 1. Discrete Variables Engr. Dr. Attaullah Shah.
Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 5-1 Chapter 5 Some Important Discrete Probability Distributions Business Statistics,
Random Variables Lecture Lecturer : FATEN AL-HUSSAIN.
AP Statistics From Randomness to Probability Chapter 14.
Dr Hidayathulla Shaikh Probability. Objectives At the end of the lecture student should be able to Define probability Explain probability Enumerate laws.
Probability Distributions ( 확률분포 ) Chapter 5. 2 모든 가능한 ( 확률 ) 변수의 값에 대해 확률을 할당하는 체계 X 가 1, 2, …, 6 의 값을 가진다면 이 6 개 변수 값에 확률을 할당하는 함수 Definition.
Theoretical distributions: the other distributions.
Biostatistics Class 2 Probability 2/1/2000.
Chapter Five The Binomial Probability Distribution and Related Topics
Discrete Random Variables
Business Statistics Topic 4
Chapter 5 Sampling Distributions
Chapter 5 Some Important Discrete Probability Distributions
Probability Probability underlies statistical inference - the drawing of conclusions from a sample of data. If samples are drawn at random, their characteristics.
Econometric Models The most basic econometric model consists of a relationship between two variables which is disturbed by a random error. We need to use.
Discrete Probability Distributions
Probability distributions
Chapter 5 Sampling Distributions
Discrete Probability Distributions
Discrete Probability Distributions
Discrete Probability Distributions
Discrete Probability Distributions
Discrete Probability Distributions
Presentation transcript:

Problem A newly married couple plans to have four children and would like to have three girls and a boy. What are the chances (probability) their desire will be fulfilled ? The probability that a person will die within a month after a heart transplant is 0.18 (18%). What are the probabilities that in three such operations one, two or all three persons will survive ?

BINOMIAL DISTRIBUTION AND ITS APPLICATION

Probability Scale Absolute certainty—1.0—( a stone thrown up will come done to earth) 0.9 0.8 0.7 0.6 Equally Likely --- 0.5--- (when a coin is tossed , the Head will come up) 0.4 0.3 0.2 0.1 Absolute Impossibility–--- 0.0– ( I will travel to the sun tomorrow)

Characteristics of Probabilities Probabilities are expressed as fractions between 0.0 and 1.0 e.g., 0.01, 0.05, 0.10, 0.50, 0.80 Probability of a certain event = 1.0 Probability of an impossible event = 0.0 Application to biomedical research e.g., ask if results of study or experiment could be due to chance alone e.g., significance level and power e.g., sensitivity, specificity, predictive values

Basic Probability Concepts Foundation of statistics because of the concept of sampling and the concept of variation or dispersion and how likely an observed difference is due to chance Probability statements used frequently in statistics e.g., we say that we are 90% sure that an observed treatment effect in a study is real

No. of survivals after operations Total no. of cases operated For example, if a surgeon performs renal transplants in 150 cases and succeeds in 63 cases then probability of survival after operation in calculated as: No. of survivals after operations p = -------------------------- Total no. of cases operated 63 = ------------------------ = 0.42 150 And chances of not surviving i.e., probability of dying cases, q or (1-p) is 1-0.42 = 0.58 which can be shown as: No. of cases died after the operation q = ------------------------------------------------------------------ Total number of cases operated 87 = ------------------------------- = 0.58

Definition of Probabilities If some process is repeated a large number of times, n, and if some resulting event with the characteristic of E occurs m times, the relative frequency of occurrence of E, m/n, will be approximately equal to the probability of E: P(E)=m/n Also known as relative frequency

Elementary Properties of Probabilities - I Probability of an event is a non-negative number Given some process (or experiment) with n mutually exclusive outcomes (events), E1, E2, …, En, the probability of any event Ei is assigned a nonnegative number P(Ei) 0

Elementary Properties of Probabilities - II Sum of the probabilities of mutually exclusive outcomes is equal to 1 Property of exhaustiveness refers to the fact that the observer of the process must allow for all possible outcomes P(E1) + P(E2) + … + P(En) = 1

Elementary Properties of Probabilities - III Probability of occurrence of either of two mutually exclusive events is equal to the sum of their individual probabilities Given two mutually exclusive events A and B P(A or B) = P(A) + P(B)

Examples: (1) Suppose we toss a die. What is the probability of 4 coming up? since there are six mutually exclusive and equally likely outcomes out of which 4 in only one, the probability of 4 coming up is 1/6. (2) Suppose we toss 2 coins. We can have the following outcomes: both heads, HH; one head and the other tail, TH or HT; and both tails, TT (H=Head; T=Tail). Suppose we want to know the probability of HH. HH being one of the four equally likely outcomes, the probability of obtaining HH is ¼.

(3) Suppose we throw 2 dice and we want the probability of a total of 7 points. A total of 7 can come in 6 ways (1-6,2-5,3-4,4-3,5-2, or 6-1). So the numerator will be 6. Since we have 6 sides for each die, the total number of ‘equally likely’ ‘mutually exclusive’ outcomes is 6 x 6 =36. So the chance of getting a total of 7 when we throw 2 dice is 6/36 (or 1/6).

Examples: (Addition Law) When we toss a die, what is the probability of getting 2 or 4 or 6 ? The prob. of 2 =1/6 The prob. of 4=1/6 The prob. of 6=1/6 Probability of 2 or 4 or 6 is: 1/6 +1/6+1/6 = 3/6 = ½ (Multiplication Law) In tossing 2 coins, Prob. of head in one coin =1/2 Prob. of head in another coin=1/2 Thus prob. of head in both coins =1/2 x 1/2 =1/4

Combinations Based on last example, it is clear that we need to calculate more easily the probability of a particular result If a set consists of n objects, and we wish to form a subset of x objects from these n objects, without regard to order of the objects in the subset, the result is called a combination The number of combinations of n objects taken x at a time is given by nCk = n! / (k! ( n-k)!) Where k! (factorial) is the product of all numbers from k to 0 0! = 1

Permutations Similar to combinations If a set consists of n objects, and we wish to form a subset of x objects from these n objects, taking into account the order of the objects in the subset, the result is called a permutation The number of permutations of n objects taken x at a time is given by nPk = n! / ( n-k)!

Probability distributions of discrete variables A table, graph, formula, or other device used to specify all possible values of a discrete random variable along with their respective probabilities P(X=x) Tables – value, frequency, probability Graph – usually bar chart Formula - Binomial distribution

Theoretical Probability Distributions -- If we know (reasonably) that data are from a certain distribution, than we know a lot about it -- Means, standard deviations, other measures of dispersion That knowledge makes it easier to make statistical inference; i.e., to test differences Many types of distributions 1300+ have been documented in the literature Three main ones Binomial (discrete - 0,1) Poisson (discrete counts) Normal (continuous)

Binomial Distribution Derived from a series of binary outcomes called a Bernoulli trial When a random process or experiment, called a trial, can result in only one of two mutually exclusive outcomes, such as dead or alive, sick or well, the trial is called a Bernoulli trial

Bernoulli Process A sequence of Bernoulli trials forms a Bernoulli process under the following conditions Each trial results in one of two possible, mutually exclusive, outcomes: “success” and “failure” Probability of success, p, remains constant from trial to trial. Probability of failure is q = 1-p. Trials are independent; that is, success in one trial does not influence the probability of success in a subsequent trial.

Bernoulli Process - Example Probability of a certain sequence of binary outcomes (Bernoulli trials) is a function of p and q. For example, a particular sequence of 3 “successes” and 2 “failures” can be represented by p*p*p*q*q; = p3q2 However, if we ask for the probability of 3 “successes” and 2 “failures” in a set of 5 trials, then we need to know how may possible combinations of 3 successes and 2 failures out of all of the possible outcomes there are.

Binomial Distribution The binomial probability density function f(x) = nCx px qn-x for x=0,1,2,3…,n = n! / (k! ( n-k)!) px qn-x This is called the binomial distribution

Parameters of Binomial distribution n (the number of objects) and p (the probability of a ‘success’) are the two unknown quantities which define a binomial distribution. They are called the parameters of the binomial distribution. The Binomial distribution is applicable to the situation where: the n trails are independent (ie., what occurs in one trail does not affect what will occur in the next trail), at each trail there is only two possible outcomes (‘success’ or ‘failure’) the probability of a ‘success’ (p) should be known and is the same for all trails

Mean and Variance of the Binomial distribution To obtain the frequency distribution for a particular combination of ’ n ‘and ‘p ‘we need to calculate the probabilities p (X=0), p (X=2), ------, p (X=n) Where X is the random variable representing the number of ‘successes’ in ‘n’ trails. Where ‘n’ is large, the calculations of these probabilities becomes very tedious and time consuming. However, we can obtain the mean and variance of ‘X’ as a summary of the distribution. For a binomial distribution mean ( ) is given by ‘np’ and the variance (2 ) is equal to ‘ np(1-p)’. The mean is the average value of the random variable that would be expected to occur in the long run and the standard deviation is the expected variation from the mean

Binomial Table Normally, we would look up probabilities in the Binomial Table Tables of the Binomial probability distribution function P (X=k) Find probability that x=4 successes when n trials = 10 and p of success = 0.3 Find probability that x4 Find probability that x5

In medical research, an outcome of interest can often be expressed as the presence or absence of a particular disease, sign or symptom or as whether the patient lived or died, or recovered or did not recover. In each case we are dealing with an outcome in which exactly one of two alternatives can occur.

Suppose we know that the survival rate for a particular disease is 20% and we have 10 patients with this disease. We would use the binomial distribution to calculate the probability of having 3 or fewer patients survive. The answer is 0.88, so that we have about a 90% chance of having 3 or fewer patients surviving (or 7 or more dying)

(d) P( two or fewer) = P(X<=2) =1-0.729 =0.271 P( two or three) = P(X=2 or X=3) =P(X=2) +P(X=3) =0.972 (f) P( exactly three) = P(X=3) = 0.729