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1. Welcome T0 A Session On Probability

2. Your Text Book Bussiness Statisties By S.P.Gupta &M. P. Gupta (Fourteenth Enlarged edition) Chapter 11

3. What is Probability? It is a value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur in the future. Example: If a company has only five sales regions, and each region’s name or number is written on a slip of paper and the slip put in a hat, the probability of selecting one of the five regions is 1. The probability of selecting from that hat a slip paper that reads “Pittsburgh steelers” is 0. Thus, the probability of 1 represents something that is certain to happen, and the probability of 0 represents something that cannot happen. The closer a probability is to 0, the more improbable it is the event will happen.

4. What is the use of Probability theory? As there is uncertainty in decision making, it is important to evaluate scientifically known risks involved . For this evaluation, probability theory (which is often referred to as the science of uncertainty) is helpful. The use of probability theory allows the decision maker with only limited information to analyze the risks and minimize the gamble inherent. For example, in marketing a new product or accepting an incoming shipment possibly a shipment contains defective parts.

5. How is probability frequently expressed? It is frequently expressed as a decimal, such as .70, .27, .50 etc. It may also be given as a fraction such as It can assume any number form 0 to 1, inclusive.

6. What are the key words used in the study of probability? The three key words which are used in probability are experiment, outcome and event. Experiment: It is a process that leads to the occurrence of one and only one of several possible observations. [ In probability, an experiment has two or more possible results, and it is uncertain which will occur. Outcome: A particular result of an experiment. Event: A collection of one or more outcomes of an experiment. Clarification of the definitions of Experiment, Outcome and event.

7. Experiment Rolling of die All possible outcomes Observe a 1 Observe a 2 Observe a 3 Observe a 4 Observe a 5 Observe a 6 Some possible events. Observe an even number Observe a number greater than 4 Observe a number 3 or less.

8. Example: • Consider the experiment of tossing two coins once • The sample Space S = {H H , HT, TH, TT} • Consider the event of one head • Probability of one head =

9. What are the approaches to assigning probabilities? • The approaches to assigning probabilities given are : • Objective probability • Subjective probability Continued……

10. What are the approaches to assigning Probabilities? Objective probability Classical probability Empirical probability

11. What is classical probability? It is based on the assumption that the outcomes of an experiment are equally likely. Using this classical view point, classical probability is defined as follows: Probability of an event =

12. Example: Consider an experiment of rolling a six-sided die. What is the probability of the event “an even number of spots appear face up”? The possible outcomes are Continued……

13. There are three favorable outcomes (a two, a four, and a six ) in the collection of six equally likely possible outcomes. • Probability of an event = Continued…….

14. What is mutually exclusive event? • The occurrence of one event means that none of the other events can occur at the same time. • Example : • A manufactured part can be either acceptable or unacceptable, but cannot be both at the same time. In a sample of manufactured parts, the event of selecting an acceptable part and the event of selecting an unacceptable part are mutually exclusive. • An employee selected at random is either male or female but cannot be both. So these two events are mutually exclusive. Continued…….

15. Collectively exhaustive events At least one of the events must occur when an experiment is conducted. Example: If an experiment has a set of events that includes every possible outcome, such as the events “ an even number” and “ an odd number” in the die-tossing experiment, then the set of events is “collectively exhaustive.’’ For the die- tossing experiment, every outcome will be either even or odd. So the set is collectively exhaustive. Continued……..

16. Sum of probabilities If the set of events is collectively exhaustive and the events are mutually exclusive, the sum of the probabilities is 1.

17. Relative frequency concept (empirical probability) The probability of an event happening in the long run is determined by observing what fraction of the time similar events happened in the past: Probability of event =

18. Example: On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 113 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed? Let P (A) stand for the probability a future mission is successfully completed. Probability of successful flight =  the probability that a future mission is successful is

19. Example: Throughout her career professor Jones has awarded 186 A’s out of the 1200 students she has taught. What is the probability that a student in her section this semester with receive an A? Probability of receiving an A=

20. What is subjective probability? If there is little or no past experience or information on which to base a probability, it may be arrived at subjectively. This means an individual evaluates the available opinions and other information and then estimates or assigns the probability. This probability is aptly called a subjective probability. Subjective concept of probability. This refers to the likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.

21. Examples: • Estimating the likelihood the New England Patriots will play in the Super Bowl next Year • Estimating the probability General Motors Corp. will lose its number1 ranking in total units sold to Ford Motor Co. or Daimler Chrysler within 2 years. • Estimating the likelihood you will earn an A in this course.

22. What are the various types of probabilities?

23. Example: Approaches to Probability Objective Subjective Classical Probability Empirical Probability Based on equally likely outcomes Based on relative frequencies

24. What are the basic rules of probability? • If events are mutually exclusive, then the occurrence of any one of the events precludes any of the other events from occurring. • Rules of addition: If two events A and B are mutually exclusive, the special rule of addition states that the probability of A or B occurring equals the sum of their probabilities. • This rule is expressed in the following formula: • P (A or B) = P(A) + P(B). • For three mutually exclusive events designated A, B, and c, the rule is written: • P (A or B or c) = P(A) +P (B) + P ( c ).

25. Example : An automatic machine fills plastic bags with a mixture of bears, broccoli and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the bears and other vegetables, a package might be underweight or overweight. A check of 4, 000 packages filled in the past month revealed: Weight Underweight Satisfactory Overweight Number of packages 100 3,600 300 A B C Total = 4000

26. What is the probability that a particular package will be either underweight or overweight? Example: Probability of underweight, Probability of overweight, Note: Here, the events are mutually exclusive. It means that a package of mixed vegetables cannot be underweight, satisfactory, and overweight at the same time. They are also collectively exhaustive, that is, a selected passage must be either underweight, satisfactory, or overweight.

27. What are the complement rules? • The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. If P(A) is the probability of event A and P(~A) is the complement of A, P(A) + P(~A) = 1 OR P(A) = 1-P(~A).

28. What are the complement rules? A Venn Diagram illustrating the complement rule would appear as: ~A A

29. Example 3 New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York:

30. Example 4 • Recall Example 3. • If C is the event that a flight arrives on time, then P(C) = 800/1000 = .8. • If D is the event that a flight is canceled, then P(D) = 25/1000 = .025. • Use the complement rule to show that the probability of an early (A) or a late (B) flight is .175.

31. Example 4 P(A or B) = 1 - P(C or D) = 1 -[.8 +.025] =.175 D .025 C .8 ~(C or D) = (A or B) .175

32. A sample of employees of worldwide enterprises is to be surveyed about a new pension plan. The employees are classified as follows: Classification Supervisors Maintenance Production Management Secretarial Event A B C D E Number of Employees 120 50 1,460 302 68

33. a ) What is the probability that the first person is i. either in maintenance or a secretariate? ii. not in management? b) Are the events in part (a) (i) complementary or mutually exclusive or both?

34. Total number of employees = 2000 • Probability that the first person selected in maintenance • Probability that the first person selected is in secretarial (E), •  Probability that the first person selected is either in maintenance or secretarial = P (B or E) Continued…..

35. ii) Probability that the first person is in management, • By applying complement rule, we can get probability that the first person is not in management. • i.e. P(~D) = C) They are not complementary, but mutually exclusive. Continued…..

36. Example 5 In a sample of 500 students, 320 said they had a stereo, 175 said they had a TV, and 100 said they had both: TV 175 Stereo 320 Both 100

37. Example 5 • If a student is selected at random, what is the probability that the student has only a stereo, only a TV, and both a stereo and TV? P(S) = 320/500 = .64. P(T) = 175/500 = .35. P(S and T) = 100/500 =.20. • If a student is selected at random, what is the probability that the student has either a stereo or a TV in his or her room? P(S or T) = P(S) + P(T) - P(S and T) = .64 +.35 -.20 = .79.

38. What is joint probability? Joint Probability is a probability that measures the likelihood that two or more events will happen concurrently. An example would be the event that a student has both a stereo and TV in his or her room.

39. What are the special rules of multiplications? • The special rule of multiplication requires that two events A and B are independent. • Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. The special rule is written: P(A and B) = P(A)*P(B).

40. Example 6 Chris owns two stocks which are independent of each other. The probability that stock A increases in value next year is .5. The probability that stock B will increase in value next year is .7. • What is the probability that both stocks increase in value next year? P(A and B) = (.5)(.7) = .35. • What is the probability that at least one of these stocks increase in value during the next year (this implies that either one can increase or both)? Thus, P(at least one) = (.5)(.3) + (.5)(.7) + (.7)(.5) = .85.

41. What is conditional probability? Conditional probability is the probability of a particular event occurring, given that another event has occurred. Note: The probability of the event A, given that the event B has occurred, is denoted by P(A|B).

42. What are the general multiplication rules? The general rule of multiplication is used to find the joint probability that two events will occur, as it states: for two events A and B, the joint probability that both events will happen is found by multiplying the probability that event A will happen by the conditional probability of B given that A has occurred. • The joint probability, P(A and B) is given by the following formula:P(A and B) = P(A)*P(B|A) Or P(A and B) = P(B)*P(A|B)

43. Exercise 1 The Dean of the School of Business at Miami collected the following information about undergraduate students in her college:

44. In a lottery game, three numbers are randomly selected from a tumbler of balls numbered 1 through 50. • a) How many permutations are possible? • b) How many combinations are possible? • Answers. The number of permutation possible Continued……

45. The number of combination possible is:

46. 2. Bayes’ Theorem

48. Bayes’ theorem

49. Example : The Betts Machine shop, Inc., has 8 screw machines but only three spaces available in the production area for the machines. In how many different ways can the eight machines be arranged in the three spaces available? Answer: Here, number of machines = 8 Spaces available = 3

50. Example : Three electronic parts are to be assembled into a plug–in unit for a television set. The parts can be assembled in any order. In how many different ways can the three parts be assembled? Answer: Continued…..