Sample Spaces Collection of all possible outcomes e.g.: All six faces of a dice: e.g.: All 52 cards in a deck:

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

Visualizing Events Contingency tables Tree diagrams Black 2 24 26 Ace Not Ace Total Black 2 24 26 Red 2 24 26 Total 4 48 52 Ace Red Cards Not an Ace Full Deck of Cards Ace Black Cards Not an Ace

Simple and Joint Events The Event of a Triangle The event of a triangle AND blue in color There are 5 triangles in this collection of 18 objects Two triangles that are blue

 Special Events Null Event 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 

Special Events Mutually exclusive 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

Probability Sample of 1,000 households in terms of purchase behaviour for big-screen TV sets.

Decision Tree

Sample of 300 households whether the TV set purchased was an HDTV and whether they also purchased a DVD player. Draw the decision tree for purchased a DVD player and an HDTV.

Joint Probability Using Contingency Table Event Event B1 B2 Total A1 P(A1 and B1) P(A1 and B2) P(A1) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) P(B2) 1 Marginal (Simple) Probability Joint Probability

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

Compound Probability (Addition Rule) P(A1 or B1 ) = P(A1) + P(B1) - P(A1 and B1) Event Event B1 B2 Total A1 P(A1 and B1) P(A1 and B2) P(A1) A2 P(A2 and B1) P(A2 and B2) P(A2) Total P(B1) P(B2) 1 For Mutually Exclusive Events: P(A or B) = P(A) + P(B)

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

Conditional Probability and Statistical Independence Multiplication rule:

Conditional Probability and Statistical Independence 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

Bayes’s Theorem Adding up the parts of A in all the B’s Same Event

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?

Bayes’s Theorem Using Contingency Table Repay Repay Total College .2 .05 .25 .3 .45 .75 College Total .5 .5 1.0