1. 1 1 Slide Introduction to Probability Assigning Probabilities and Probability Relationships Chapter 4 BA 201

2. 2 2 Slide ASSIGNING PROBABILITIES

3. 3 3 Slide Assigning Probabilities Basic Requirements for Assigning Probabilities Basic Requirements for Assigning Probabilities 1. The probability assigned to each experimental outcome must be between 0 and 1, inclusively. 0 < P ( E i ) < 1 for all i where: E i is the i th experimental outcome and P ( E i ) is its probability

4. 4 4 Slide Assigning Probabilities Basic Requirements for Assigning Probabilities Basic Requirements for Assigning Probabilities 2. The sum of the probabilities for all experimental outcomes must equal 1. P ( E 1 ) + P ( E 2 ) +... + P ( E n ) = 1 where: n is the number of experimental outcomes

5. 5 5 Slide Assigning Probabilities Classical Method Relative Frequency Method Subjective Method Assigning probabilities based on the assumption of equally likely outcomes Assigning probabilities based on experimentation or historical data Assigning probabilities based on judgment

6. 6 6 Slide Classical Method If an experiment has n possible outcomes, the classical method would assign a probability of 1/ n to each outcome. Experiment: Rolling a die Sample Space: S = {1, 2, 3, 4, 5, 6} Probabilities: Each sample point has a 1/6 chance of occurring 1/6+1/6+1/6+1/6+1/6+1/6 = 1 Example: Rolling a Die Example: Rolling a Die

7. 7 7 Slide Relative Frequency Method Number of Polishers Rented Number of Days 0123401234 4 6 18 10 2 Lucas Tool Rental would like to assign probabilities to the number of car polishers it rents each day. Office records show the following frequencies of daily rentals for the last 40 days. Lucas Tool Rental

8. 8 8 Slide Each probability assignment is given by dividing the frequency (number of days) by the total frequency (total number of days). Relative Frequency Method 4/404/40 Probability Number of Polishers Rented Number of Days 0123401234 4 6 18 10 2 40 0.10 0.15 0.45 0.25 0.05 1.00 Example: Lucas Tool Rental Example: Lucas Tool Rental

9. 9 9 Slide Subjective Method When economic conditions and a company’s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data. We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur. The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate.

10. 10 Slide Subjective Method An analyst made the following probability estimates. Exper. Outcome Net Gain or LossProbability (10, 8) (10,  2) (5, 8) (5,  2) (0, 8) (0,  2) (  20, 8) (  20,  2) $18,000 Gain $8,000 Gain $13,000 Gain $3,000 Gain $8,000 Gain $2,000 Loss $12,000 Loss $22,000 Loss 0.20 0.08 0.16 0.26 0.10 0.12 0.02 0.06 Example: Bradley Investments Example: Bradley Investments

11. 11 Slide An event is a collection of sample points. The probability of any event is equal to the sum of the probabilities of the sample points in the event. If we can identify all the sample points of an experiment and assign a probability to each, we can compute the probability of an event. Events and Their Probabilities

12. 12 Slide Events and Their Probabilities Event M = Markley Oil Profitable M = {(10, 8), (10,  2), (5, 8), (5,  2)} P ( M ) = P (10, 8) + P (10,  2) + P (5, 8) + P (5,  2) = 0.20 + 0.08 + 0.16 + 0.26.0= 0.70.0= 0.70 Bradley Investments

13. 13 Slide Events and Their Probabilities Event C = Collins Mining Profitable C = {(10, 8), (5, 8), (0, 8), (  20, 8)} P ( C ) = P (10, 8) + P (5, 8) + P (0, 8) + P (  20, 8) = 0.20 + 0.16 + 0.10 + 0.02 = 0.48 Example: Bradley Investments Example: Bradley Investments

14. 14 Slide PROBABILITY RELATIONSHIPS

15. 15 Slide Some Basic Relationships of Probability There are some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities. Complement of an Event Intersection of Two Events Mutually Exclusive Events Union of Two Events

16. 16 Slide The complement of A is denoted by A c. The complement of event A is defined to be the event consisting of all sample points that are not in A. Complement of an Event Event A AcAc Sample Space S

17. 17 Slide The union of events A and B is denoted by A  B  The union of events A and B is the event containing all sample points that are in A or B or both. Union of Two Events Sample Space S Event A Event B

18. 18 Slide Union of Two Events Event M = Markley Oil Profitable Event C = Collins Mining Profitable M  C = Markley Oil Profitable or Collins Mining Profitable (or both) M  C = {(10, 8), (10,  2), (5, 8), (5,  2), (0, 8), (  20, 8)} P ( M  C) = P (10, 8) + P (10,  2) + P (5, 8) + P (5,  2) + P (0, 8) + P (  20, 8) = 0.20 + 0.08 + 0.16 + 0.26 + 0.10 + 0.02 = 0.82 Example: Bradley Investments Example: Bradley Investments

19. 19 Slide The intersection of events A and B is denoted by A  The intersection of events A and B is the set of all sample points that are in both A and B. Sample Space S Event A Event B Intersection of Two Events Intersection of A and B

20. 20 Slide Intersection of Two Events Event M = Markley Oil Profitable Event C = Collins Mining Profitable M  C = Markley Oil Profitable and Collins Mining Profitable M  C = {(10, 8), (5, 8)} P ( M  C) = P (10, 8) + P (5, 8) = 0.20 + 0.16 = 0.36 Bradley Investments

21. 21 Slide Mutually Exclusive Events Two events are said to be mutually exclusive if the events have no sample points in common. Two events are mutually exclusive if, when one event occurs, the other cannot occur. Sample Space S Sample Space S Event A Event B

22. 22 Slide Mutually Exclusive Events If events A and B are mutually exclusive, P ( A  B  = 0. The addition law for mutually exclusive events is: P ( A  B ) = P ( A ) + P ( B )

23. 23 Slide PRACTICE PROBABILITY RELATIONSHIPS

24. 24 Slide Practice Based on the Venn diagram, describe the following: A B C

25. 25 Slide Practice Out of forty students, 14 are taking English and 29 are taking Chemistry. Five students are in both classes. Draw a Venn Diagram depicting this relationship.

26. 26 Slide Practice Students are either undergraduate students or graduate students. Draw a Venn Diagram depicting this relationship.

27. 27 Slide Scenario OutcomeProbability O1O1 0.10 O2O2 0.30 O3O3 0.05 O4O4 0.15 O5O5 0.20 O6O6 0.05 O7O7 0.10 O8O8 0.05 EventOutcomes E1E1 O 1, O 3 E2E2 O 1, O 4, O 5, O 6 E3E3 O2O2 E4E4 O 7, O 8 E5E5 O 4, O 5, O 7 E6E6 O 1, O 7 A statistical experiment has the following outcomes, along with their probabilities and the following events, with the corresponding outcomes. Outcomes Events

28. 28 Slide Probabilities of Events What is the probability of E 1 [P(E 1 )]?

29. 29 Slide Probabilities of Events What is the probability of E 2 [P(E 2 )]?

30. 30 Slide Probabilities of Events What is the probability of E 3 [P(E 3 )]?

31. 31 Slide Probabilities of Events What outcomes are in the complement of E 2 [E 2 c ]?

32. 32 Slide Probabilities of Events What outcomes are in the union of E 3 and E 4 ? (E 3 ⋃ E 4 )? What is the probability of E 3 ⋃ E 4 [P(E 3 ⋃ E 4 )]?

33. 33 Slide Probabilities of Events What outcomes are in the intersection of E 2 and E 5 (E 2 ⋂ E 5 )? What is the probability of E 2 ⋂ E 5 [P(E 2 ⋂ E 5 )]?

34. 34 Slide Probabilities of Events Based on the available information, are E 2 and E 4 mutually exclusive? What is the probability of E 2 ⋂ E 4 [P(E 2 ⋂ E 4 )]? What is the probability of E 2 ⋃ E 4 [P(E 2 ⋃ E 4 )]?

35. 35 Slide