1. BA 201 Lecture 6 Basic Probability Concepts

2. 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

3. Population and Sample PopulationSample Use parameters to summarize features Use statistics to summarize features Inference on the population from the sample p.??

4. Approaches to Probability A priori 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.155

5. 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

6. 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.156

7. Special Events 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   Null Event p.159

8. Special Events 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 (continued) p.159

9. 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. ??-??

10. Using Contingency Table (Joint Probability Table, Venn Diagram) Attribute A Attribute B pp. ??-?? (continued)

11. Using Contingency Table (Joint Probability Table, Venn Diagram) Attribute A Attribute B Total Joint probabilities Marginal probabilities pp. ??-?? (continued)

12. Using Contingency Table (Joint Probability Table, Venn Diagram) 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. ??-?? (continued)

13. Using Contingency Table (Joint Probability Table, Venn Diagram) 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? pp. ??-?? (continued)

14. Computing Joint Probability: The Multiplication Rule The Probability of a Joint Event, A and B: p.170

15. Computing Compound Probability: The Addition Rule Probability of a Compound Event, A or B: p.160

16. Conditional Probability Conditional Probability: p.166

17. 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. ??-??

18. Bayes’ Theorem Using Contingency Table (continued) Attribute A Attribute B pp. ??-??

19. Bayes’ Theorem Using the Formula p.175

20. 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 p.169

21. 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