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# Chapter 1 probability

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### Chapter 1 probability

1. 1. Basic Terminology:-• Trial & Event• Exhaustive number of cases• Favorable number of cases• Mutually Exclusive events• Independent events
2. 2. • Classical Method:- Assigning probabilities based on the assumption of equally likely outcomes. favorable no. of cases probabilit y of an event Exhaustive no. of cases
3. 3. • Relative Frequency Method:- Assigning probabilities based on experimentation or historical data.
4. 4. Probability rules:-1. The case where one event or another event occur.2. The case where two or more events will both occur.
5. 5. Ex:-Union shop steward Peter has drafted a set of wage and benefit demands to be presented to management. To get an idea of worker support for the package, he randomly polls the two largest groups of workers at his plant, The machinists (M) and the inspectors (I). He polls 30 of each group with the following results: Opinion of package M I Strongly support 9 10 Mildly support 11 3 Undecided 2 2 Mildly oppose 4 8 Strongly oppose 4 7i. What is the pbt that a machinist randomly selected from the polled group mildly supports the package?ii. What is the pbt that an inspector randomly selected from the polled group is undecided about the package?iii. What is the pbt that a worker (machinist or inspector) randomly selected from the polled group strongly or mildly supports the package?iv. What types of pbts are these?
6. 6. Addition rule:If A and B are any two events and are disjoint then P(A or B)= The Pbt of either A or B happening = P(A)+P(B)
7. 7. If A and B are any two events and are not disjoint then P(A or B)= The Pbt of either A or B happening = P(A)+P(B)-P(AB)P(A) is that Pbt of A happeningP(B) is that Pbt of B happeningP(AB) is that Pbt of A and B happening together
8. 8. Ex: A leading marketing research firm in India, wants to collect information about households with computers and Internet access in urban Mumbai. After conducting an intensive survey, it was revealed that 60% of the households have computers with Internet access; 70% of the households have two or more computer sets. Suppose 50% of the households have computers with Internet connection and two or more computers. A household with computer is randomly selected.
9. 9. 1. What is the pbt that the household has computers with Internet access or two or more computers?2. What is the pbt that the household has computers with Internet access or two or more computers but not both?3. What is the pbt that the household has neither computers with Internet access nor two or more computers?
10. 10. Ex: A company is interested in understanding the consumer behavior of the capital of the newly formed state Chattisgarh, Raipur. For this purpose, a company has selected a sample of 300 consumers and asked a question, “Do you enjoy shopping?” Out of 300 respondents 200 were males, 100 were females. Out of 200 males, 120 responded “yes” and out of 100 females, 70 responded “yes”.A respondent is selected randomly. Construct a pbt matrix and find the following pbts:
11. 11. • The respondent is a male• Enjoy shopping• Is a female and enjoys shopping• Is a male and does not enjoy shopping• Is a female or enjoys shopping• Is a male or does not enjoy shopping• Is a male or female
12. 12. 1. The employees of a certain company have elected 5 of their number to represent them on the employee-management productivity council. Profiles of the 5 are as follows: Gender Age 1. Male 30 2. Male 32 3. Female 45 4. Female 20 5. Male 40 This group decides to elect a spokesperson by drawing a name from a chit. What is the Pbt the spokesperson will be either female or over 35?2. An inspector of the Alaska pipeline has the task of comparing the reliability of two pumping stations. Each station is susceptible to two kinds of failure: pump failure and leakage. When either (or both) occur, the station must be shut down. The data at hand indicate that the following pbts prevail: Station P (Pump failure) P (Leakage) P (Both) 1 0.07 0.10 0 2 0.09 0.12 0.06 Which station has the higher pbt of being shut down?
13. 13. Pbts under conditions of Statistical independence:1. Marginal / Unconditional pbts2. Joint pbts3. Conditional pbts
14. 14. If A & B are two independent events then the Joint pbt of A and B is given by P(A and B)=P(AB)=P(A)P(B)If A & B are two independent events then the conditional pbt of B given that A is given by P(B/A)=P(B) Similarly the conditional pbt of A given that B is given by P(A/B)=P(A)
15. 15. 1.A bag contains 32 marbles:4 are red, 9 are black, 12 are blue, 6 are yellow, and 1 is purple. Marbles are drawn one at a time with replacement. What is the pbt that i. The second marble is yellow given the first was yellow? ii. The second marble is yellow given the first was black? iii. The third marble is purple given both the first and second were purple?
16. 16. 2. The health dept. routinely conducts two independent inspections of each restaurant, with the restaurant passing only if both inspectors pass it. Inspector A is very experienced, and hence passes only 2% of restaurants that actually do have health code violations. Inspector B is less experienced and passes 7% of restaurants with violations. What is the pbt that a). Inspector A passes a restaurant with a violation, given that inspector B has found a violation? b). Inspector B passes a restaurant with a violation, given that inspector A passes it? c). A restaurant with a violation is passed by the health department?
17. 17. 3. Unique Pvt. Ltd is a company involved in the production of small bearings. One day an important machine stops working. The company has three operators. Their chances of repairing machine are: ½,1/3,1/4 respectively. What is the probability that the machine will be repaired when they try independently?
18. 18. Pbts under conditions of Statistical dependence:1. Conditional pbts: If A and B are any two events then the conditional pbt of B given that already the event A happened is given by P(AB) P(B/A) P(A) Similarly the conditional pbt of A given that already the event B happened is given by P(AB) P(A/B) P(B)
19. 19. 2. Joint pbts: (Multiplication law of pbt) If A and B are any two events then P(A and B)=P(AB)=P(A)P(B/A) =P(B)P(A/B)
20. 20. Let a box contains 10 balls distributed as below• Three are colored and dotted• One is colored and striped• Two are gray and dotted• Four are gray and striped a) Suppose a ball drawn from the box and found to be colored. What is the pbt that it is dotted? b) Suppose a ball drawn from the box and found to be colored. What is the pbt that it is striped? c) Suppose a ball drawn from the box and found to be gray. What is the pbt that it is dotted? d) Suppose a ball drawn from the box and found to be gray. What is the pbt that it is striped?
21. 21. d) Suppose a ball is drawn from thebox, find the pbt that it is colored andsriped? e) Suppose a ball is drawn from thebox, find the pbt that it is colored?
22. 22. 1.Two events A and B are statistically dependent. If P(A)=0.39, P(B)=0.21and P(A or B)=0.47, find the pbt that a). Neither A nor B will occur b). Both A & B will occur c). B will occur, given that A has occurred. d). A will occur, given that B has occurred
23. 23. 2. During a study of auto accidents, the Highway safety found that 60% of all accidents occur at night. 52% are alcohol- related, and 37% occur at night and are alcohol- related. a) What is the pbt that an accident was alcohol-related, given that it occurred at night? b) What is the pbt that an accident occurred at night, given that it was alcohol- related?
24. 24. 3. The university’s library has been randomly surveying patrons over the last month to see who is using the library and what services they have been using. Patrons are classified as undergraduate, graduate, or faculty. Services are classified as reference, periodicals, or books. The data for 350 people are given below. Assume a patron uses only one service per visit. Patron Reference Periodicals Books Undergraduate 44 26 72 Graduate 24 61 20 Faculty 16 69 18 Find the pbt that a randomly selected chosen patron a) Is a graduate student b) Visited the periodicals section, given that the patron is a graduate c) Is a faculty member, given a reference section visit. d) Is an undergraduate who visited the book section.
25. 25. 4. The southeast regional manager of General Express, a private parcel – delivery firm, is worried about the likelihood of strikes by some of his employees. He has learned that the chance of a strike by his pilots is 0.75 and the chance of a strike by his drivers is 0.65. Further, he knows that if the drivers strike , there is a 90% chance that the pilots will strike in sympathy. a) What is the pbt of both group’s striking? b) If the pilots strike, what is the pbt that the drivers will strike in sympathy?
26. 26. Posterior probabilities or Bayes’ theorem:-• Often we begin probability analysis with initial or prior probabilities.• Then, from a sample, special report, or a product test we obtain some additional information.• Given this information, we calculate revised or posterior probabilities.• Bayes’ theorem provides the means for revising the prior probabilities. Application Prior New Posterior of Bayes’ Probabilities Information Probabilities Theorem
27. 27. 1.In a bolt factory machines A, B and C manufactures respectively 25%,35% and 40% of the total output. Of their output 5%, 4%,2% are defective bolts. A bolt is drawn from the output and is found to be defective. What is the chance that it was produced by machine B?
28. 28. Let E1 be the event of drawing a bolt at random manufactured by the machine A E2 be the event of drawing a bolt at random manufactured by the machine B E3 be the event of drawing a bolt at random manufactured by the machine C
29. 29. Let X be the event of its being defective
30. 30. Prior pbts: P(E1)=25% P(E2)=35% P(E3)=40%Likelihood pbts: P(X/E1)=5% P(X/E2)=4% P(X/E3)=2%
31. 31. Additional information: A defective bolt was selected from the output.
32. 32. To find the chance that it was produced by machine B we apply the Bayes’ theotem and is given by P( Ei )P( X | Ei ) P( Ei | X ) P( E1 )P( X | E1 ) P( E2 )P( X | E2 ) ... P( En )P( X | En )
33. 33. P(E 2 )P(X | E 2 )P(E 2 | X) P(E1 )P(X | E1 )P(E 2 )P(X | E 2 )P(E 3 )P(X | E 3 ) (0.35)(0.04) 140 28 0.4058 (0.25)(0.05) (0.35)(0.04) (0.40)(0.02) 345 69
34. 34. Prior PosteriorEvents Probabilities Conditional Joint Probabilities Probabilities Ei P(Ei) Probabilities P(X|Ei) P(Ei ∩ X) P(Ei |X) E1 0.25 0.05 0.0125 0.3623 E2 0.35 0.04 0.014 0.4058 E3 0.40 0.02 0.008 0.2319
35. 35. 1. T.C.Fox, marketing director for Metro-Goldmine Motion Pictures, believe that the studio’s upcoming release has a 60% chance of being a hit, a 25% chance of being a moderate success, and a 15% chance of being a flop. To test the accuracy of his opinion, T.C.Fox has scheduled two test screenings. After each screening, the audience rates the film on scale of 1 to 10, 10 being best. From his long experience in the industry, T.C. Fox knows that 60% of the time a hit picture will receive a rating of 7 or higher; 30% of the time, it will receive a rating of 4,5, or6; and 10% of the time, it will receive a rating of 3 or lower. For moderately successful picture, the respective pbts are 0.30, 0.45, and 0.25; for a flop film, respective pbts are 0.15, 0.35, and 0.50.a) If the first test screening produces a score of 6, what is the pbt that the film will be a hit?b) If the first test screening produces a score of 6 and the second test screening yields a score of 2, what is the pbt that the film will be a flop assuming that the screening results are independent of each other?
36. 36. Case Let:A state Democratic official has decided that changes in the state unemployment rate will have a major effect on his party’s chance of gaining or losing seats in the state senate. He has determined that if unemployment rises by 2% or more, the respective pbts of losing more than 10 seats, losing 6 to 10 seats, gaining or losing 5 or less seats, gaining 6 to 10 seats , and gaining more than 10 seats are 0.25, 0.35, 0.15, 0.15, and 0.10 . If unemployment changes by less than 2% , the respective pbts of losing more than 10 seats, losing 6 to 10 seats, gaining or losing 5 or less seats, gaining 6 to 10 seats , and gaining more than 10 seats are 0.10, 0.10, 0.15, 0.35, and 0.30. If unemployment falls by 2% or more, the respective pbts of losing more than 10 seats, losing 6 to 10 seats, gaining or losing 5 or less seats, gaining 6 to 10 seats , and gaining more than 10 seats are 0.05, 0.10, 0.10, 0.40, and 0.35. Currently this official believes that unemployment will by 2% or more with pbt 0.25, change by less than 2% with pbt 0.45, and fall by 2% or more with pbt 0.30. a) If the Democrats gained seven seats, what is the pbt that unemployment fell by 2% or more? b) If the Democrats lost one lost seat, what is the pbt that unemployment changed by less than 2%?