Basic Business Statistics 8 th Edition Chapter 4

Chapter Topics n Basic probability concepts n n Conditional probability n n Sample spaces and events, simple probability, joint probability Statistical independence, marginal probability Bayes’s theorem © 2002 Prentice-Hall, Inc. 2

Sample Spaces n Collection of all possible outcomes n e. g. : All six

Events n Simple event n n n Outcome from a sample space with one characteristic e. g. : A red card from a deck of cards Joint event n n Involves two outcomes simultaneously e. g. : An ace that is also red from a deck of cards © 2002 Prentice-Hall, Inc. 4

Visualizing Events n Contingency tables Ace Black Red Total n Tree diagrams Full Deck

Simple Events The Event of a Triangle There are 5 triangles in this collection

Joint Events The event of a triangle AND blue in color Two triangles that

Special Events Null Event n Impossible event e. g. : Club & diamond on one card draw n Complement of event n n n © 2002 Prentice-Hall, Inc. 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 8

Special Events n (continued) Mutually exclusive events n n 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 n n n 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 n Events B, C and D are also collectively exhaustive n © 2002 Prentice-Hall, Inc. 9

Contingency Table A Deck of 52 Cards Red Ace Not an Ace Total Red

Tree Diagram Event Possibilities Full Deck of Cards © 2002 Prentice-Hall, Inc. Red Cards

Probability n n n Probability is the numerical measure of the likelihood that an event will occur 1 Certain Value is between 0 and 1 Sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1 © 2002 Prentice-Hall, Inc. . 5 0 Impossible 12

Computing Probabilities n The probability of an event E: e. g. P( ) = 2/36 (There are 2 ways to get one 6 and the other 4) n Each of the outcomes in the sample space is equally likely to occur © 2002 Prentice-Hall, Inc. 13

Computing Joint Probability n The probability of a joint event, A and B: ©

Joint Probability Using Contingency Table Event B 1 Event Total A 1 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 Joint Probability © 2002 Prentice-Hall, Inc. B 2 P(B 1) P(B 2) 1 Marginal (Simple) Probability 15

Computing Compound Probability n Probability of a compound event, A or B: © 2002

Compound Probability (Addition Rule) P(A 1 or B 1 ) = P(A 1) + P(B 1) - P(A 1 and B 1) Event B 1 B 2 Total A 1 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 For Mutually Exclusive Events: P(A or B) = P(A) + P(B) © 2002 Prentice-Hall, Inc. 17

Computing Conditional Probability n The probability of event A given that event B has

Conditional Probability Using Contingency Table Type Color Red Black Total Ace 2 2 4

Conditional Probability and Statistical Independence n Conditional probability: n Multiplication rule: © 2002 Prentice-Hall,

Conditional Probability and Statistical Independence (continued) n n Events A and B are independent if Events A and B are independent when the probability of one event, A, is not affected by another event, B © 2002 Prentice-Hall, Inc. 21

Bayes’s Theorem Same Event © 2002 Prentice-Hall, Inc. Adding up the parts of A

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? © 2002 Prentice-Hall, Inc. 23

Bayes’s Theorem Using Contingency Table (continued) © 2002 Prentice-Hall, Inc. Repay Total College .

Chapter Summary n Discussed basic probability concepts n n Defined conditional probability n n Sample spaces and events, simple probability, and joint probability Statistical independence, marginal probability Discussed Bayes’s theorem © 2002 Prentice-Hall, Inc. 25