Basic Business Statistics 8 th Edition Chapter 4
Basic Business Statistics (8 th Edition) Chapter 4 Basic Probability © 2002 Prentice-Hall, Inc. Chap 4 -1
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 faces of a die: n e. g. : All 52 cards in a deck: © 2002 Prentice-Hall, Inc. 3
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 of Cards © 2002 Prentice-Hall, Inc. Red Cards Black Cards 2 2 4 Not Ace 24 24 48 Total 26 26 52 Ace Not an Ace 5
Simple Events The Event of a Triangle There are 5 triangles in this collection of 18 objects © 2002 Prentice-Hall, Inc. 6
Joint Events The event of a triangle AND blue in color Two triangles that are blue © 2002 Prentice-Hall, Inc. 7
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 2 24 26 Black 2 24 26 Total 4 48 52 Sample Space © 2002 Prentice-Hall, Inc. 10
Tree Diagram Event Possibilities Full Deck of Cards © 2002 Prentice-Hall, Inc. Red Cards Ace Not an Ace Black Cards Not an Ace 11
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: © 2002 Prentice-Hall, Inc. 14
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 Prentice-Hall, Inc. 16
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 occurred: © 2002 Prentice-Hall, Inc. 18
Conditional Probability Using Contingency Table Type Color Red Black Total Ace 2 2 4 Non-Ace 24 24 48 Total 26 26 52 Revised Sample Space © 2002 Prentice-Hall, Inc. 19
Conditional Probability and Statistical Independence n Conditional probability: n Multiplication rule: © 2002 Prentice-Hall, Inc. 20
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 in all the B’s 22
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 . 2 . 05 . 25 College . 3 . 45 . 75 Total . 5 1. 0 24
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
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