Chapter 3: Probability Concepts and Distributions PDF

Chapter 3: Probability Concepts and Distributions Statistics, Data Analysis, and Decision Modeling, Fifth Edition James R. Evans Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-1...

Chapter 3: Probability Concepts and Distributions Statistics, Data Analysis, and Decision Modeling, Fifth Edition James R. Evans Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-1  Probability Probability – the likelihood that an outcome occurs Probabilities are values between 0 and 1. The closer the probability is to 1, the more likely it is that the outcome will occur. Some convert probabilities to percentages The statement “there is a 10% chance that oil prices will rise next quarter” is another way of stating that “the probability of a rise in oil prices is 0.1.” Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-2  Experiments and Outcomes Experiment – a process that results in some outcome Roll dice Observe and record weather conditions Conduct market survey Watch the stock market Outcome – an observed result of an experiment Sum of the dice Description of the weather Proportion of respondents who favor a product Change in the Dow Jones Industrial Average Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-3  Sample Space Sample space - all possible outcomes of an experiment Dice rolls: 2, 3, …, 12 Weather outcomes: clear, partly cloudy, cloudy Customer reaction: proportion who favor a product (a number between 0 and 1) Change in DJIA: positive or negative real number Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-4  Three Views of Probability Classical definition: based on theory Relative frequency: based on empirical data Subjective: based on judgment Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-5  Classical Definition Probability = number of favorable outcomes divided by the total number of possible outcomes Example: There are six ways of rolling a 7 with a pair of dice, and 36 possible rolls. Therefore, the probability of rolling a 7 is 6/36 = 0.167. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-6  Relative Frequency Definition Probability = number of times an event has occurred in the past divided by the total number of observations Example: Of the last 10 days when certain weather conditions have been observed, it has rained the next day 8 times. The probability of rain the next day is 0.80 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-7  Subjective Definition What is the probability that the New York Yankees will win the World Series this year? What is the probability your school will win its conference championship this year? What is the probability the NASDAQ will go up 2% next week? Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-8  Basic Probability Rules and Formulas 1. Probability associated with any outcome must be between 0 and 1 0 ≤ P(Oi) ≤ 1 for each outcome Oi 2. Sum of probabilities over all possible outcomes must be 1.0 P(O1) + P(O2) + … + P(On) = 1 Example: Flip a coin three times Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT Each has probability of (1/2)3 = 1/8 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-9  Events An event is a collection of one or more outcomes from Obtaining a 7 or 11 on a roll of dice Having a clear or partly cloudy day The proportion of respondents that favor a product is at least 0.60 Having a positive weekly change in the Dow If A is any event, the complement of A, denoted Ac, consists of all outcomes in the sample space not in A. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-10  Rule 1 The probability of any event is the sum of the probabilities of the outcomes that compose that event. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-11  Rule 2 The probability of the complement of any event A is P(Ac) = 1 - P(A). Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-12  Mutually Exclusive Events Two events are mutually exclusive if they have no outcomes in common. A B B C B&C A and B are mutually exclusive B and C are not Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-13  Rule 3 If events A and B are mutually exclusive, then P(A or B) = P(A) + P(B). Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-14  Rule 4 If two events A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) - P(A and B). Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-15  Example What is the probability of obtaining exactly two heads or exactly two tails in 3 flips of a coin? These events are mutually exclusive. Probability = 3/8 + 3/8 = 6/8 What is the probability of obtaining at least two tails or at least one head? A = {TTT, TTH, THT, HTT}, B = {TTH, THT, THH, HTT, HTH, HHT, HHH} The events are not mutually exclusive. P(A) = 4/8; P(B) = 7/8; P(A&B) = 3/8. Therefore, P(A or B) = 4/8 + 7/8 – 3/8 = 1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-16  Conditional Probability Conditional probability – the probability of the occurrence of one event, A, given that another event B is known to be true or have already occurred. P(A|B) = P(A and B)/P(B) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-17  Example Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-18  Multiplication Law of Probability P(A and B) = P(A |B) P(B) = P(B |A) P(A) Two events A and B are independent if P(A |B) = P(A) If A and B are independent, then P(A and B) = P(B)P(A) = P(A)P(B) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-19  Random Variables Random variable – a numerical description of the outcome of an experiment. Random variables are denoted by capital letters, X, Y, …; specific values by lower case letters, x, y, … Discrete random variable – the number of possible outcomes can be counted Continuous random variable – outcomes over one or more continuous intervals of real numbers Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-20  Examples of Random Variables Experiment: flip a coin 3 times. Outcomes: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH Random variable: X = number of heads. X can be either 0, 1, 2, or 3. Experiment: observe end-of-week closing stock price. Random variable: Y = closing stock price. X can be any nonnegative real number. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-21  Probability Distributions Probability distribution – a characterization of the possible values a random variable may assume along with the probability of assuming these values. Probability distributions may be defined for both discrete and continuous random variables. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-22  Example: Theoretical Probability Distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-23  Example: Empirical Probability Distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-24  Example: Subjective Probability Distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-25  Discrete Random Variables Probability mass function f(x): specifies the probability of each discrete outcome Two properties: Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-26  Cumulative Distribution Function, F(x) Specifies the probability that the random variable X will be less than or equal to x, denoted as P(X ≤ x). Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-27  Expected Value and Variance of Discrete Random Variables Expected value of a random variable X is the theoretical analogy of the mean, or weighted average of possible values:  E[X] =  x i f(x i ) i =1 Variance and standard deviation of a random variable X:  Var[X] =  (x j - E[X])2 f(x j ) j=1  X =  j (x j=1 - E[X]) 2 f(x j ) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-28  Example You play a lottery in which you buy a ticket for $50 and are told you have a 1 in 1000 chance of winning $25,000. The random variable X is your net winnings, and its probability distribution is x f(x) -$50 0.999 $24,950 0.001 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-29  Calculations E[X] = -$50(0.999) + $24,950(0.001) = -$25.00 Var[X] = (-50 - [-25.00])2(0.999) + (24,950 - [-25.00])2(0.001) = 624,375 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-30  Discrete Probability Distributions Bernoulli Binomial Poisson Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-31  Bernoulli Distribution A random variable with two possible outcomes: “success” (x = 0) and “failure” (x = 1) p = probability of “success” Expected value = p; variance = p(1 – p) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-32  Binomial Distribution n independent replications of a Bernoulli experiment, each with constant probability of success p X represents the number of successes in n experiments. Expected value =np Variance =np(1-p) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-33  Examples of the Binomial Distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-34  Excel Function BINOM.DIST(number_s, trials, probability_s, cumulative) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-35  Poisson Distribution Models the number of occurrences in some unit of measure, e.g., events per unit time, number of items per order X = number of events that occur; x = 0, 1, 2, Expected value = ; variance =  Poisson approximates binomial when n is large and p small Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-36  Example of the Poisson Distribution ( = 12) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-37  Excel Function POISSON.DIST(x, mean, cumulative) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-38  Continuous Probability Distributions Probability density function, f(x), a continuous function that describes the probability of outcomes for a continuous random variable X. A histogram of sample data approximates the shape of the underlying density function. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-39  Refining Subjective Probabilities Toward a Continuous Distribution Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-40  Properties of Probability Density Functions f(x)  0 for all x Total area under f(x) = 1 There are always infinitely many values for X P(X = x) = 0 We can only define probabilities over intervals: P(a ≤ X ≤ b), P(X < c), or P(X > d) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-41  Cumulative Distribution Function F(x) specifies the probability that the random variable X will be less than or equal to x; that is, P(X  x). F(x) is equal to the area under f(x) to the left of x The probability that X is between a and b is the area under f(x) from a to b: P(a ≤ X ≤ b) = P(X ≤b) - P(X ≤ a) = F(b) – F(a) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-42  Properties of Continuous Distributions Continuous distributions have one or more parameters that characterize the density function: Shape parameter – controls the shape of the distribution Scale parameter – controls the unit of measurement Location parameter – specifies the location relative to zero on the horizontal axis Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-43  Uniform Distribution EV[X] = (a + b)/2 V[X] = (b – a)2/12 a = location b = scale for fixed a Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-44  Normal Distribution Familiar bell-shaped curve. Symmetric, median = mean = mode; half the area is on either side of the mean Range is unbounded: the curve never touches the x-axis Parameters Mean,  (location) Variance 2 > 0 (scale) Density function: -(x-  ) 2 e 2 2 f(x) = 2 2 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-45  Standard Normal Distribution Standard normal: mean = 0, variance = 1, denoted as N(0,1) See Appendix Table A.1 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-46  Excel Functions NORM.DIST(x, mean, standard_deviation, cumulative ) NORM.DIST(x, mean, standard_deviation, TRUE) calculates the cumulative probability F(x) = P(X ≤ x) for a specified mean and standard deviation. NORM.S.DIST(z) NORM.S.DIST(z) generates the cumulative probability for a standard normal distribution. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-47  Example Customer demand (X) is normal with a mean of 750 units/month and a standard deviation of 100 units/month. 1. What is the probability that demand will be at most 900 units? 2. What is the probability that demand will exceed 700 units? 3. What is the probability that demand will be between 700 and 900 units? 4. What level of demand would be exceeded at most 10% of the time? Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-48  Excel Probability Tabulation Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-49  P(X>900) NORM.DIST(900,750,100,TR UE) = 0.9332. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-50  P(X>700) P(X >700) = 1 - P(X < 700) = 1 – F (700) = 1 - 0.3085 = 0.6915 =1- NORM.DIST(700,750,100,TRUE) Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-51  P(700 0.10 NORM.INV(0.90,750,100) = 878.155 x = 878 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-53  Standard Normal Probabilities Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-54  Triangular Distribution Three parameters: Minimum, a Maximum, b Most likely, c a is the location parameter; b is the scale parameter for fixed a; c the shape parameter. EV[X] = (a + b + c)/3 V[X] = (a2 + b2 + c2 – ab – ac – bc)/18 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-55  Exponential Distribution Models events that occur randomly over time Customer arrivals, machine failures Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-56  Properties of Exponential Excel function EXPONDIST(x, lambda, cumulative). EV[X] = 1/ V[X] = 1/2 Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-57  Other Useful Distributions Lognormal Negative Binomial Gamma Hypergeometric Weibull/Erlang Logistic Beta Pareto Geometric Extreme value Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-58  Joint and Marginal Probability Distributions Joint probability distribution – A probability distribution that specifies the probabilities of outcomes of two different random variables, X and Y, that occur at the same time, or jointly Marginal probability – probability associated with the outcomes of each random variable regardless of the value of the other Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-59  Example Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-60  Binomial Distribution Theory Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-61  Poisson Distribution Theory Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-62  Uniform Distribution Theory Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-63  Normal Distribution Theory Standardized normal values (z- values): Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-64  Exponential Distribution Theory Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-65  All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright ©2013 Pearson Education, Inc. publishing as Prentice Hall 3-66