BA 201 Lecture 6 Basic Probability Concepts
Topics • Basic Probability Concepts • Approaches to probability • Sample spaces • Events and special events • Using Contingency Table (Joint Probability Table, Venn Diagram) • The multiplication rule • The addition rule • Conditional probability • The Bayes’ theorem • Statistical Independence
Population and Sample p.?? Population Sample Use statistics to summarize features Use parameters to summarize features Inference on the population from the sample
p.155 Approaches to Probability • Apriori classical probability • Based on prior knowledge of the process involved • E.g. Analyze the scenarios when tossing a fair coin • Empirical classical probability • Based on observed data • E.g. Record the number of heads and tails in repeated trials of tossing a coin • Subjective probability • Based on individual’s past experience, personal opinion and analysis of a particular situation • E.g. Evaluate the status of a coin someone has offered to use to gamble with
p.156 Sample Spaces • Collection of All Possible Outcomes • E.g. All 6 faces of a die: • E.g. All 52 cards of a bridge deck:
p.156 Events • Simple Event • Outcome from a sample space with 1 characteristic • E.g. A Red Card from a deck of cards • Joint Event • Involves 2 outcomes simultaneously • E.g. An Ace which is also a Red Card from a deck of cards
p.159 Special Events Null Event • Impossible Event • Impossible event • E.g. Club & Diamond on 1 card draw • Complement of Event • For event A, all events not in A • Denoted as A’ • E.g. A: Queen of Diamond A’: All cards in a deck that are not Queen of Diamond
p.159 Special Events (continued) • Mutually Exclusive Events • Two events cannot occur together • E.g. A: Queen of Diamond; B: Queen of Club • 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
pp. ??-?? Using Contingency Table (Joint Probability Table, Venn Diagram) 50% of borrowers repaid their student loans. 20% of the borrowers were students who had a college degree and repaid their loans. 25% of the students earned a college degree.
pp. ??-?? Using Contingency Table (Joint Probability Table, Venn Diagram) (continued) Attribute A AttributeB
pp. ??-?? Using Contingency Table (Joint Probability Table, Venn Diagram) (continued) Attribute A AttributeB Total Total Joint probabilities Marginal probabilities
pp. ??-?? Using Contingency Table (Joint Probability Table, Venn Diagram) (continued) What is the probability that a randomly selected borrower will have a college degree or repay the loan? What is the probability that a randomly selected borrower will have a college degree and default on the loan?
pp. ??-?? Using Contingency Table (Joint Probability Table, Venn Diagram) (continued) If you randomly select a borrower and have found out that he/she has defaulted on the loan, what is the probability that he/she has a college degree? If you randomly select a borrower and have found out that he/she does not have a college degree, what is the probability that he/she will default?
p.170 Computing Joint Probability:The Multiplication Rule • The Probability of a Joint Event, A and B:
p.160 Computing Compound Probability: The Addition Rule • Probability of a Compound Event, A or B:
p.166 Conditional Probability • Conditional Probability:
pp. ??-?? Bayes’ Theorem Using Contingency Table 50% of borrowers repaid their loans. Out of those who repaid, 40% had a college degree. 10% of those who defaulted had a college degree. What is the probability that a randomly selected borrow who has a college degree will repay the loan?
pp. ??-?? Bayes’ Theorem Using Contingency Table (continued) Attribute A AttributeB
p.175 Bayes’ Theorem Using the Formula
p.169 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
Summary • Introduced Basic Probability Concepts • Approaches to probability • Sample spaces • Events and special events • Illustrated Using Contingency Table (Joint Probability Table, Venn Diagram) • The multiplication rule • The addition rule • Conditional probability • The Bayes’ theorem • Discussed Statistical Independence
