1. 1 STAT 500 – Statistics for Managers STAT 500 Statistics for Managers

2. 2 STAT 500 – Statistics for Managers Objectives for this Session Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

3. 3 STAT 500 – Statistics for Managers Objectives for this Session Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

4. 4 STAT 500 – Statistics for Managers Probability Probability is a statement (of degree of belief) that expresses the chance that some event will occur. Experiment: An activity or measurement that results in an outcome. Sample Space: All possible outcomes of an experiment. Event: One or more of the possible outcomes of an experiment; a subset of the sample space.

5. 5 STAT 500 – Statistics for Managers Objectives for this Session#2 Part 1 Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

6. 6 STAT 500 – Statistics for Managers Probabilities  Stands for summation

7. 7 STAT 500 – Statistics for Managers Probability Distribution A probability distribution describes all the possible outcomes of a random experiment. Consider for example, tossing a coin. P (Head) = 0.5 P (Tail) = 0.5

8. 8 STAT 500 – Statistics for Managers Probability of an Event The probability of event A equals the sum of the probabilities of the individual outcomes that constitute event A, and it is denoted by P(A). The event consisting of all outcomes not included in event A is called the complement of event A. It is denoted by A c.

9. 9 STAT 500 – Statistics for Managers Union and Intersection of Events The event of all outcomes that are in event A, in event B, or in both events, is called the union of events A and B. It’s denoted by A  B. The event consisting of all outcomes that are in both event A and in event B is called the intersection of events A and B. It is denoted by A  B.

10. 10 STAT 500 – Statistics for Managers Venn Diagrams Event A C shaded. A

11. 11 STAT 500 – Statistics for Managers Venn Diagrams Events A  B shaded. A B

12. 12 STAT 500 – Statistics for Managers Rules of Probability For any event A, P(A) + P(A c ) = 1 For any events A and B, P(A  B) = P(A) + P(B) - P(A  B) Events are called mutually exclusive if they cannot occur simultaneously. Events A and B are mutually exclusive if A  B is the empty or null set and is denoted by {  }.

13. 13 STAT 500 – Statistics for Managers Objectives for this Session#2 Part 1 Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

14. 14 STAT 500 – Statistics for Managers Conditional Probability The conditional probability of event B given that event A has occurred is denoted by P(B|A) and is p (B | A) = p (A  B) / p (A), and similarly, p (A | B) = p (A  B) / p (B),

15. 15 STAT 500 – Statistics for Managers Independent Events When the conditional probability of event B given that event A has occurred equals the unconditional probability of event A, events A and B are called independent events. If p(B | A) = p (B) then A and B are independent events. p(A  B) = p(A) * p(B), when A and B are independent.

16. 16 STAT 500 – Statistics for Managers Objectives for this Session#2 Part 1 Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

17. 17 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Many companies in the rapidly growing high- technology field are seeking to diversify their manufacturing base by locating branch facilities in small but progressive communities. In this way they can develop new sources of labor and increase their rate of retention of key management personnel. Suppose a preliminary analysis by an electronics firm has identified 5 communities that meet organizational objectives for site location.

18. 18 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Of the 5 communities, 2 are located in Virginia state and 3 others are not. The President of the company plans to select two communities at random for in-depth discussions concerning the location of a manufacturing facility. We are interested in the geographic location of each chosen community.

19. 19 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Question # 1 Define the experiment Selection of two communities at random out of five communities. Question # 2 List the sample space Let C = {C1, C2, C3, C4, C5} be the set of 5 communities. Sample space consists of all possible outcomes of the experiment, i.e., O = {C1C2, C1C3, C1C4, C1C5, C2C3, C2C4, C2C5, C3C4, C3C5, C4C5}

20. 20 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Question # 3 If all pairs of communities have an equal chance of selection, what is the probability that the two Virginia communities will be chosen by the president. The sample space consists of ten outcomes: O = {C1C2, C1C3, C1C4, C1C5, C2C3, C2C4, C2C5, C3C4, C3C5, C4C5} Event: Both the communities are from Virginia p(Event) = 1/10 = 0.1

21. 21 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Question # 4 What is the probability that the community chosen by the President will consist of at least one from Virginia. The sample space consists of ten outcomes: O = {C1C2, C1C3, C1C4, C1C5, C2C3, C2C4, C2C5, C3C4, C3C5, C4C5} Event: At least one community is from Virginia p(Event) = 7/10 = 0.7

22. 22 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Question # 5 What is the probability that the community chosen by the President will consist of none from Virginia. The sample space consists of ten outcomes: O = {C1C2, C1C3, C1C4, C1C5, C2C3, C2C4, C2C5, C3C4, C3C5, C4C5} Event: At least one community is from Virginia p(Event) = 7/10 = 0.7 p(none from Virginia) = 1 – 0.7 = 0.3

23. 23 STAT 500 – Statistics for Managers Sample Space / Equally Likely / Complement Suppose the closing price of IBM stock at New York Stock Exchange is $150. Describe your beliefs about the stock price for the next day.. Up. Down. No Change * Up by > $ 10 * Up <= $10 * No Change * Down > $ 10 * Down <= $10

24. 24 STAT 500 – Statistics for Managers Objectives for this Session#2 Part 1 Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

25. 25 STAT 500 – Statistics for Managers Probability A sample of 300 customers of America Online resulted in the frequency distribution of monthly charges: What is the probability that the monthly charges are $100 or more? What is the probability that the monthly charges are less than $ 100?

26. 26 STAT 500 – Statistics for Managers Probability The probability that the monthly charges are $100 or more is = 0.63 (99+87+3)/300 The probability that the monthly charges are less than $ 100 is = 0.37 (42+69) / 300

27. 27 STAT 500 – Statistics for Managers Objectives for this Session#2 Part 1 Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3

28. 28 STAT 500 – Statistics for Managers Probability The manager of a local hotel chain projects the probability estimates of vacancies for the next weekend. What is the probability that there are no vacancies? What is the probability that there are at least 3 vacancies? What is the probability that there are at most 2 vacancies?

29. 29 STAT 500 – Statistics for Managers Probability What is the probability that there are no vacancies? P(no Vacancies) = 0.25 What is the probability that there are at least 3 vacancies? P(at least 3 vacancies) = P(3) + P(4) = 0.30 What is the probability that there are at most 2 vacancies? P(0) + p(1) + p(2) = 0.7

30. 30 STAT 500 – Statistics for Managers Objectives for this Session#2 Part 1 Probability – basic terms Properties of probability Conditional probability Example # 1 Sample Space Example # 2 Simple illustration Example # 3