Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions 2002 Prentice-Hall, Inc. Chap 4-1
Chapter Topics Basic probability concepts Sample spaces and events, simple probability, joint probability Conditional probability Statistical independence, marginal probability Bayes s Theorem 2002 Prentice-Hall, Inc. Chap 4-2
Chapter Topics (continued) The probability of a discrete random variable Covariance and its applications in finance Binomial distribution Poisson distribution Hypergeometric distribution 2002 Prentice-Hall, Inc. Chap 4-3
Sample Spaces Collection of all possible outcomes e.g.: All six faces of a die: e.g.: All 52 cards in a deck: 2002 Prentice-Hall, Inc. Chap 4-4
Events Simple event Outcome from a sample space with one characteristic e.g.: A red card from a deck of cards Joint event Involves two outcomes simultaneously e.g.: An ace that is also red from a deck of cards 2002 Prentice-Hall, Inc. Chap 4-5
Visualizing Events Contingency Tables Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Tree Diagrams Full Deck of Cards Red Cards Black Cards Ace Not an Ace Ace Not an Ace 2002 Prentice-Hall, Inc. Chap 4-6
The Event of a Triangle Simple Events There are 5 triangles in this collection of 18 objects 2002 Prentice-Hall, Inc. Chap 4-7
Joint Events The event of a triangle AND blue in color Two triangles that are blue 2002 Prentice-Hall, Inc. Chap 4-8
Special Events Impossible event e.g.: Club & diamond on one card draw Complement of event For event A, all events not in A Denoted as A e.g.: A: queen of diamonds A : all cards in a deck that are not queen of diamonds Null Event 2002 Prentice-Hall, Inc. Chap 4-9
Mutually exclusive events Special Events Two events cannot occur together e.g.: -- A: queen of diamonds; B: queen of clubs Events A and B are mutually exclusive Collectively exhaustive events One of the events must occur The set of events covers the whole sample space e.g.: -- A: all the aces; B: all the black cards; C: all the diamonds; D: all the hearts Events A, B, C and D are collectively exhaustive Events B, C and D are also collectively exhaustive (continued) 2002 Prentice-Hall, Inc. Chap 4-10
Contingency Table Red Ace Red Black Total A Deck of 52 Cards Ace Not an Ace 2 24 2 24 Total 2002 Prentice-Hall, Inc. Chap 4-11 26 26 4 48 52 Sample Space
Tree Diagram Event Possibilities Full Deck of Cards Red Cards Black Cards Ace Not an Ace Ace Not an Ace 2002 Prentice-Hall, Inc. Chap 4-12
Probability Probability is the numerical measure of the likelihood that an event will occur Value is between 0 and 1 Sum of the probabilities of all mutually exclusive and collective exhaustive events is 1 1.5 0 Certain Impossible 2002 Prentice-Hall, Inc. Chap 4-13
Computing Probabilities PE ( ) The probability of an event E: number of event outcomes total number of possible outcomes in the sample space X T e.g. P( ) = 2/36 (There are 2 ways to get one 6 and the other 4) Each of the outcomes in the sample space is equally likely to occur 2002 Prentice-Hall, Inc. Chap 4-14
Computing Joint Probability The probability of a joint event, A and B: P( A and B) = P( A B) number of outcomes from both A and B total number of possible outcomes in sample space E.g. P(Red Card and Ace) 2 Red Aces 1 52 Total Number of Cards 26 2002 Prentice-Hall, Inc. Chap 4-15
Joint Probability Using Contingency Table Event A 1 Event B 1 B 2 Total P(A 1 and B 1 ) P(A 1 and B 2 ) P(A 1 ) A 2 P(A 2 and B 1 ) P(A 2 and B 2 ) P(A 2 ) Total P(B 1 ) P(B 2 ) 1 Joint Probability Marginal (Simple) Probability 2002 Prentice-Hall, Inc. Chap 4-16
Computing Compound Probability Probability of a compound event, A or B: P( A or B) P( AB) E.g. number of outcomes from either A or B or both total number of outcomes in sample space P(Red Card or Ace) 4 Aces + 26 Red Cards - 2 Red Aces 52 total number of cards 28 7 52 13 2002 Prentice-Hall, Inc. Chap 4-17
Compound Probability (Addition Rule) P(A 1 or B 1 ) = P(A 1 ) + P(B 1 ) - P(A 1 and B 1 ) Event A 1 Event B 1 B 2 P(A 1 and B 1 ) P(A 1 and B 2 ) Total P(A 1 ) A 2 P(A 2 and B 1 ) P(A 2 and B 2 ) P(A 2 ) Total P(B 1 ) P(B 2 ) 1 For Mutually Exclusive Events: P(A or B) = P(A) + P(B) 2002 Prentice-Hall, Inc. Chap 4-18
Computing Conditional Probability The probability of event A given that event B has occurred: P( A B) E.g. P( A and B) PB ( ) P(Red Card given that it is an Ace) 2 Red Aces 1 4 Aces 2 2002 Prentice-Hall, Inc. Chap 4-19
Conditional Probability Using Contingency Table Color Type Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 P(Ace Red) Revised Sample Space P(Ace and Red) 2/52 2 P(Red) 26/ 52 26 2002 Prentice-Hall, Inc. Chap 4-20
Conditional Probability and Statistical Independence Conditional probability: P( A B) P( A and B) PB ( ) Multiplication rule: P( A and B) P( A B) P( B) P( B A) P( A) 2002 Prentice-Hall, Inc. Chap 4-21
Conditional Probability and Statistical Independence Events A and B are independent if P( A B) P( A) or P( B A) P( B) or P( A and B) P( A) P( B) (continued) Events A and B are independent when the probability of one event, A, is not affected by another event, B 2002 Prentice-Hall, Inc. Chap 4-22
Bayes s Theorem P B i A Same Event 1 1 P A B P A B P B P A B P B P B i and P A A i P B i k Adding up the parts of A in all the B s k 2002 Prentice-Hall, Inc. Chap 4-23
Bayes s Theorem Using Contingency Table Fifty percent of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. Ten percent of those who defaulted had a college degree. What is the probability that a randomly selected borrower who has a college degree will repay the loan? P R.50 P C R.4 P C R.10 PR C? 2002 Prentice-Hall, Inc. Chap 4-24
Bayes s Theorem Using Contingency Table (continued) Repay Repay Total College.2.05.25 P R C College.3.45.75 Total.5.5 1.0 P C R P R P C R P R PC R P R.4.5.2.8.4.5.1.5.25 2002 Prentice-Hall, Inc. Chap 4-25
Random Variable Random Variable Outcomes of an experiment expressed numerically e.g.: Toss a die twice; count the number of times the number 4 appears (0, 1 or 2 times) 2002 Prentice-Hall, Inc. Chap 4-26
Discrete Random Variable Discrete random variable Obtained by counting (1, 2, 3, etc.) Usually a finite number of different values e.g.: Toss a coin five times; count the number of tails (0, 1, 2, 3, 4, or 5 times) 2002 Prentice-Hall, Inc. Chap 4-27
Discrete Probability Distribution Example Event: Toss two coins Count the number of tails T T Probability Distribution Values Probability 0 1/4 =.25 1 2/4 =.50 2 1/4 =.25 T T 2002 Prentice-Hall, Inc. Chap 4-28
Discrete Probability Distribution List of all possible [X j, p(x j ) ] pairs X j = value of random variable P(X j ) = probability associated with value Mutually exclusive (nothing in common) Collectively exhaustive (nothing left out) j P X j 0 P X 1 1 2002 Prentice-Hall, Inc. Chap 4-29
Summary Measures Expected value (the mean) Weighted average of the probability distribution X jp X j E X j e.g.: Toss 2 coins, count the number of tails, compute expected value j j X P X j 0 2.5 1.5 2.25 1 2002 Prentice-Hall, Inc. Chap 4-30
Summary Measures (continued) Variance Weight average squared deviation about the mean E X X P X 2 j j 2 e.g. Toss two coins, count number of tails, compute variance 2 X j P X j 2 2 2 2 0 1.25 11.5 2 1.25.5 2 2002 Prentice-Hall, Inc. Chap 4-31
Covariance and its Application N X E X Y E Y P X Y XY i i i i i1 X : discrete random variable th X i : i outcome of X Y : discrete random variable th Yi : i outcome of Y P X Y i i : probability of occurrence of the outcome of X and the i 2002 Prentice-Hall, Inc. Chap 4-32 th i th outcome of Y
Computing the Mean for Investment Returns Return per $1,000 for two types of investments Investment P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession -$100 -$200.5 Stable Economy + 100 + 50.3 Expanding Economy + 250 + 350 E X X 100.2 100.5 250.3 $105 EY Y 200.2 50.5 350.3 $90 2002 Prentice-Hall, Inc. Chap 4-33
Computing the Variance for Investment Returns Investment P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession -$100 -$200.5 Stable Economy + 100 + 50.3 Expanding Economy + 250 + 350 2 X 2 2 2 100 105.2 100 105.5 250 105.3 14, 725 121.35 2 Y 2002 Prentice-Hall, Inc. Chap 4-34 X 2 2 2 200 90.2 50 90.5 350 90.3 37,900 194.68 Y
Computing the Covariance for Investment Returns Investment P(X i Y i ) Economic condition Dow Jones fund X Growth Stock Y.2 Recession -$100 -$200.5 Stable Economy + 100 + 50.3 Expanding Economy + 250 + 350 100 105 200 90.2 100 105 50 90.5 XY 250 105 350 90.3 23,300 The Covariance of 23,000 indicates that the two investments are positively related and will vary together in the same direction. 2002 Prentice-Hall, Inc. Chap 4-35
Important Discrete Probability Distributions Discrete Probability Distributions Binomial Hypergeometric Poisson 2002 Prentice-Hall, Inc. Chap 4-36
Binomial Probability Distribution n identical trials e.g.: 15 tosses of a coin; ten light bulbs taken from a warehouse Two mutually exclusive outcomes on each trials e.g.: Head or tail in each toss of a coin; defective or not defective light bulb Trials are independent The outcome of one trial does not affect the outcome of the other 2002 Prentice-Hall, Inc. Chap 4-37
Binomial Probability Distribution Constant probability for each trial e.g.: Probability of getting a tail is the same each time we toss the coin Two sampling methods Infinite population without replacement Finite population with replacement (continued) 2002 Prentice-Hall, Inc. Chap 4-38
Binomial Probability Distribution Function n! X P X p 1 p X! n X! P X : probability of X successes given n and p X : number of "successes" in sample X 0,1,, n p : the probability of each "success" n : sample size Tails in 2 Tosses of Coin X P(X) 0 1/4 =.25 1 2/4 =.50 2 1/4 =.25 2002 Prentice-Hall, Inc. Chap 4-39 nx
Binomial Distribution Characteristics Mean E X np E.g. np Variance and Standard Deviation 2 np 1 p E.g. np 1 5.1.5 p P(X).6.4.2 0 n = 5 p = 0.1 0 1 2 3 4 5 np 1 p 5.1 1.1.6708 X 2002 Prentice-Hall, Inc. Chap 4-40
Binomial Distribution in PHStat PHStat probability & prob. Distributions binomial Example in excel spreadsheet 2002 Prentice-Hall, Inc. Chap 4-41
Poisson Distribution Poisson Process: Discrete events in an interval The probability of One Success in an interval is stable The probability of More than One Success in this interval is 0 The probability of success is independent from interval to interval P ( X x e - x! e.g.: number of customers arriving in 15 minutes e.g.: number of defects per case of light bulbs x 2002 Prentice-Hall, Inc. Chap 4-42
P X P X X Poisson Probability Distribution Function : number of "successes" per unit : expected (average) number of "successes" e X e X! : probability of X "successes" given : 2.71828 (base of natural logs) e.g.: Find the probability of 4 customers arriving in 3 minutes when the mean is 3.6. 2002 Prentice-Hall, Inc. Chap 4-43 P X e 3.6 4! 3.6 4.1912
Poisson Distribution in PHStat PHStat probability & prob. Distributions Poisson Example in excel spreadsheet 2002 Prentice-Hall, Inc. Chap 4-44
Poisson Distribution Characteristics Mean E X N Standard Deviation and Variance 2 i1 X P X i i P(X).6.4.2 0 P(X).6.4.2 0 = 0.5 0 1 2 3 4 5 = 6 0 2 4 6 8 10 X X 2002 Prentice-Hall, Inc. Chap 4-45
Hypergeometric Distribution n trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent Concerned with finding the probability of X successes in the sample where there are A successes in the population 2002 Prentice-Hall, Inc. Chap 4-46
Hypergeometric Distribution Function A N A E.g. 3 Light bulbs were X n X selected from 10. Of the 10 P X N there were 4 defective. What is the probability that 2 of the n 3 selected are defective? P X X n N A : probability that successes given,, and n : sample size N : population size A: number of "successes" in population X : number of "successes" in sample X 0,1, 2,,n 46 2 1 2.30 10 3 2002 Prentice-Hall, Inc. Chap 4-47 P
Hypergeometric Distribution Characteristics Mean E X A n N Variance and Standard Deviation 2 na N A N n N 2 na N A N n N 2 1 1 2002 Prentice-Hall, Inc. Chap 4-48 N N Finite Population Correction Factor
Hypergeometric Distribution in PHStat PHStat probability & prob. Distributions Hypergeometric Example in excel spreadsheet 2002 Prentice-Hall, Inc. Chap 4-49
Chapter Summary Discussed basic probability concepts Sample spaces and events, simple probability, and joint probability Defined conditional probability Statistical independence, marginal probability Discussed Bayes s theorem 2002 Prentice-Hall, Inc. Chap 4-50
Chapter Summary (continued) Addressed the probability of a discrete random variable Defined covariance and discussed its application in finance Discussed binomial distribution Addressed Poisson distribution Discussed hypergeometric distribution 2002 Prentice-Hall, Inc. Chap 4-51