1. Yandell – Econ 216 Chap 4-1 Chapter 4 Basic Probability

2. Yandell – Econ 216 Chap 4-2 Chapter Goals After completing this chapter, you should be able to: Explain basic probability concepts and definitions Use contingency tables to view a sample space Apply common rules of probability Determine whether events are statistically independent Compute conditional probabilities Use Bayes’ Theorem for conditional probabilities

3. Yandell – Econ 216 Chap 4-3 Important Terms Probability – the chance that an uncertain event will occur (always between 0 and 1) Experiment – a process of obtaining outcomes for uncertain events Elementary Event – the most basic outcome possible from a simple experiment Sample Space – the collection of all possible elementary outcomes

4. Yandell – Econ 216 Chap 4-4 Assessing Probability There are three approaches to assessing the probability of un uncertain event: 1. a priori classical probability 2. empirical classical probability 3. subjective probability an individual judgment or opinion about the probability of occurrence

5. Yandell – Econ 216 Chap 4-5 Sample Space The Sample Space is the collection of all possible outcomes e.g. All 6 faces of a die: e.g. All 52 cards of a bridge deck:

6. Yandell – Econ 216 Chap 4-6 Events Simple event An outcome from a sample space with one characteristic e.g., A red card from a deck of cards Complement of an event A (denoted A' ) All outcomes that are not part of event A e.g., All cards that are not diamonds Joint event Involves two or more characteristics simultaneously e.g., An ace that is also red from a deck of cards

7. Yandell – Econ 216 Chap 4-7 Visualizing Events Contingency Tables Tree Diagrams Red 2 24 26 Black 2 24 26 Total 4 48 52 Ace Not Ace Total Full Deck of 52 Cards Red Card Black Card Not an Ace Ace Not an Ace Sample Space 2 24 2 24

8. Yandell – Econ 216 Chap 4-8 Elementary Events A automobile consultant records fuel type and vehicle type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV 6 possible elementary events: e 1 Gasoline, Truck e 2 Gasoline, Car e 3 Gasoline, SUV e 4 Diesel, Truck e 5 Diesel, Car e 6 Diesel, SUV Gasoline Diesel Car Truck Car SUV e1e2e3e4e5e6e1e2e3e4e5e6

9. Yandell – Econ 216 Chap 4-9 Probability Concepts Mutually Exclusive Events If E 1 occurs, then E 2 cannot occur E 1 and E 2 have no common elements Black Cards Red Cards A card cannot be Black and Red at the same time. E1E1 E2E2

10. Yandell – Econ 216 Chap 4-10 Collectively Exhaustive Events Collectively exhaustive events One of the events must occur The set of events covers the entire sample space example: A = aces; B = black cards; C = diamonds; D = hearts Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an ace may also be a heart) Events B, C and D are collectively exhaustive and also mutually exclusive

11. Yandell – Econ 216 Chap 4-11 Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other Dependent: Occurrence of one affects the probability of the other Probability Concepts

12. Yandell – Econ 216 Chap 4-12 Independent Events E 1 = heads on one flip of fair coin E 2 = heads on second flip of same coin Result of second flip does not depend on the result of the first flip. Dependent Events E 1 = rain forecasted on the news E 2 = take umbrella to work Probability of the second event is affected by the occurrence of the first event Independent vs. Dependent Events

13. Yandell – Econ 216 Chap 4-13 Rules of Probability Rules for Individual Values and Sum Individual ValuesSum of All Values 0 ≤ P(e i ) ≤ 1 For any event e i where: T = Number of elementary events in the sample space e i = i th elementary event

14. Yandell – Econ 216 Chap 4-14 Addition Rule for Elementary Events The probability of an event E is equal to the sum of the probabilities of the elementary events forming E That is, if: E = {e 1, e 2, e 3 } then: P(E) = P(e 1 ) + P(e 2 ) + P(e 3 )

15. Yandell – Econ 216 Chap 4-15 Complement Rule The complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E'. Complement Rule: E'E' E Or,

16. Yandell – Econ 216 Chap 4-16 Addition Rule for Two Events P(A or B) = P(A) + P(B) - P(A and B) AB Don’t count common elements twice! ■ Addition Rule: AB +=

17. Yandell – Econ 216 Chap 4-17 Addition Rule Example P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace) = 26/52 + 4/52 - 2/52 = 28/52 Don’t count the two red aces twice! Black Color Type Red Total Ace 224 Non-Ace 24 48 Total 26 52

18. Yandell – Econ 216 Chap 4-18 Addition Rule for Mutually Exclusive Events If A and B are mutually exclusive, then P(A and B) = 0 So P(A or B) = P(A) + P(B) - P(A and B) = P(A) + P(B) = 0 AB if mutually exclusive

19. Yandell – Econ 216 Chap 4-19 Conditional Probability Conditional probability for any two events A, B:

20. Yandell – Econ 216 Chap 4-20 What is the probability that a car has a CD player, given that it has AC ? i.e., we want to find P(CD | AC) Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

21. Yandell – Econ 216 Chap 4-21 Conditional Probability Example No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.61.0 Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. (continued)

22. Yandell – Econ 216 Chap 4-22 Conditional Probability Example No CDCDTotal AC.2.5.7 No AC.2.1.3 Total.4.6 1.0 Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is about 28.57%. (continued)

23. Yandell – Econ 216 Chap 4-23 For Independent Events: Conditional probability for independent events A, B:

24. Yandell – Econ 216 Chap 4-24 Multiplication Rules Multiplication rule for two events A and B: Note: If A and B are independent, then and the multiplication rule simplifies to

25. Yandell – Econ 216 Chap 4-25 Tree Diagram Example Diesel P(E 2 ) = 0.2 Gasoline P(E 1 ) = 0.8 Truck: P(E 3 |E 1 ) = 0.2 Car: P(E 4 |E 1 ) = 0.5 SUV: P(E 5 |E 1 ) = 0.3 P(E 1 and E 3 ) = 0.8 x 0.2 = 0.16 P(E 1 and E 4 ) = 0.8 x 0.5 = 0.40 P(E 1 and E 5 ) = 0.8 x 0.3 = 0.24 P(E 2 and E 3 ) = 0.2 x 0.6 = 0.12 P(E 2 and E 4 ) = 0.2 x 0.1 = 0.02 P(E 3 and E 4 ) = 0.2 x 0.3 = 0.06 Truck: P(E 3 |E 2 ) = 0.6 Car: P(E 4 |E 2 ) = 0.1 SUV: P(E 5 |E 2 ) = 0.3

26. Yandell – Econ 216 Chap 4-26 Bayes’ Theorem where: E i = i th event of interest of the T possible events B = new event that might impact P(E i ) Events E 1 to E k are mutually exclusive and collectively exhaustive

27. Yandell – Econ 216 Chap 4-27 Bayes’ Theorem Example A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?

28. Yandell – Econ 216 Chap 4-28 Let S = successful well and U = unsuccessful well P(S) =.4, P(U) =.6 (prior probabilities) Define the detailed test event as D Conditional probabilities: P(D|S) =.6 P(D|U) =.2 Revised probabilities Bayes’ Theorem Example Event Prior Prob. Conditional Prob. Joint Prob. Revised Prob. S (successful).4.6.4*.6 =.24.24/.36 =.67 U (unsuccessful).6.2.6*.2 =.12.12/.36 =.33 Sum =.36 (continued)

29. Yandell – Econ 216 Chap 4-29 Given the detailed test, the revised probability of a successful well has risen to.67 from the original estimate of.4 Bayes’ Theorem Example Event Prior Prob. Conditional Prob. Joint Prob. Revised Prob. S (successful).4.6.4*.6 =.24.24/.36 =.67 U (unsuccessful).6.2.6*.2 =.12.12/.36 =.33 Sum =.36 (continued)

30. Yandell – Econ 216 Chap 4-30 Chapter Summary Discussed basic probability concepts Sample spaces and events, contingency tables, simple probability, and joint probability Examined basic probability rules General addition rule, addition rule for mutually exclusive events, rule for collectively exhaustive events Defined conditional probability Statistical independence, marginal probability, decision trees, and the multiplication rule