MTH120_Chapter5

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MTH120_Chapter5

  1. 1. 5- 1 Chapter Five A Survey of Probability Concepts GOALS When you have completed this chapter, you will be able to:ONEDefine probability.TWODescribe the classical, empirical, and subjective approaches toprobability.THREEUnderstand the terms: experiment, event, outcome, permutations, andcombinations.FOURDefine the terms: conditional probability and joint probability.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  2. 2. 5- 2 Chapter Five continued A Survey of Probability ConceptsGOALSWhen you have completed this chapter, you will be able to:FIVECalculate probabilities applying the rules of addition and therules of multiplication.SIXUse a tree diagram to organize and compute probabilities.SEVENCalculate a probability using Bayes’ theorem.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  3. 3. 5- 3 Definitions A probability is a measure of the likelihood that an event in the future will happen. It it can only assume a value between 0 and 1. A value near zero means the event is not likely to happen. A value near one means it is likely. There are three definitions of probability: classical, empirical, and subjective.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  4. 4. 5- 4 Definitions continued  The classical definition applies when there are n equally likely outcomes.  The empirical definition applies when the number of times the event happens is divided by the number of observations.  Subjective probability is based on whatever information is available.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  5. 5. 5- 5 Definitions continued An experiment is the observation of some activity or the act of taking some measurement. An outcome is the particular result of an experiment. An event is the collection of one or more outcomes of an experiment.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  6. 6. 5- 6 Mutually Exclusive Events Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time. Events are independent if the occurrence of one event does not affect the occurrence of another.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  7. 7. 5- 7 Collectively Exhaustive Events Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  8. 8. 5- 8 Example 1 A fair die is rolled once. The experiment is rolling the die. The possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. An event is the occurrence of an even number. That is, we collect the outcomes 2, 4, and 6.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  9. 9. 5- 9 EXAMPLE 2 Throughout her teaching career Professor Jones has awarded 186 A’s out of 1,200 students. What is the probability that a student in her section this semester will receive an A?  This is an example of the empirical definition of probability.  To find the probability a selected student earned an A: 186 P( A) = = 0.155 1200McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  10. 10. 5- 10 Subjective Probability Examples of subjective probability are:  estimating the probability the Washington Redskins will win the Super Bowl this year.  estimating the probability mortgage rates for home loans will top 8 percent.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  11. 11. 5- 11 Basic Rules of Probability 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 respective probabilities: P(A or B) = P(A) + P(B)McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  12. 12. 5- 12 EXAMPLE 3  New England Commuter Airways recently supplied the following information on their commuter flights from Boston to New York: Arrival Frequency Early 100 On Time 800 Late 75 Canceled 25 Total 1000McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  13. 13. 5- 13 EXAMPLE 3 continued  If A is the event that a flight arrives early, then P(A) = 100/1000 = .10. IfB is the event that a flight arrives late, then P(B) = 75/1000 = .075. The probability that a flight is either early or late is: P(A or B) = P(A) + P(B) = .10 + .075 =.175.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  14. 14. 5- 14 The Complement Rule 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).McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  15. 15. 5- 15 The Complement Rule continued A Venn diagram illustrating the complement rule would appear as: A ~AMcGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  16. 16. 5- 16 EXAMPLE 4 Recall EXAMPLE 3. Use the complement rule to find the probability of an early (A) or a late (B) flightIfC 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.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  17. 17. 5- 17 EXAMPLE 4 continued P(A or B) = 1 - P(C or D) = 1 - [.8 +.025] =.175 D C .025 .8 ~(C or D) = (A or B) .175McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  18. 18. 5- 18 The General Rule of Addition If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B)McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  19. 19. 5- 19 The General Rule of Addition The Venn Diagram illustrates this rule: B A and B AMcGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  20. 20. 5- 20 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 Both Stereo 100 320McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  21. 21. 5- 21 EXAMPLE 5 continued  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.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  22. 22. 5- 22 EXAMPLE 5 continued  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.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  23. 23. 5- 23 Joint Probability A joint probability 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 dorm room.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  24. 24. 5- 24 Special Rule of Multiplication 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. This rule is written: P(A and B) = P(A)P(B)McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  25. 25. 5- 25 EXAMPLE 6 Chris owns two stocks, IBM and General Electric (GE). The probability that IBM stock will increase in value next year is .5 and the probability that GE stock will increase in value next year is .7. Assume the two stocks are independent. What is the probability that both stocks will increase in value next year? P(IBM and GE) = (.5)(.7) = .35.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  26. 26. 5- 26 EXAMPLE 6 continued  What is the probability that at least one of these stocks increase in value during the next year? (This means that either one can increase or both.) P(at least one) = (.5)(.3) + (.5)(.7) +(.7)(.5) = .85.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  27. 27. 5- 27 Conditional Probability A conditional probability is the probability of a particular event occurring, given that another event has occurred. The probability of the event A given that the event B has occurred is written P(A|B).McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  28. 28. 5- 28 General Multiplication Rule The general rule of multiplication is used to find the joint probability that two events will occur. It states that 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.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  29. 29. 5- 29 General Multiplication Rule 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)McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  30. 30. 5- 30 EXAMPLE 7 The Dean of the School of Business at Owens University collected the following information about undergraduate students in her college: MAJOR Male Female Total Accounting 170 110 280 Finance 120 100 220 Marketing 160 70 230 Management 150 120 270 Total 600 400 1000McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  31. 31. 5- 31 EXAMPLE 7 continued If a student is selected at random, what is the probability that the student is a female (F) accounting major (A) P(A and F) = 110/1000. Given that the student is a female, what is the probability that she is an accounting major? P(A|F) = P(A and F)/P(F) = [110/1000]/[400/1000] = .275McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  32. 32. 5- 32 Tree Diagrams A tree diagram is useful for portraying conditional and joint probabilities. It is particularly useful for analyzing business decisions involving several stages. EXAMPLE 8: In a bag containing 7 red chips and 5 blue chips you select 2 chips one after the other without replacement. Construct a tree diagram showing this information.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  33. 33. 5- 33 EXAMPLE 8 continued 6/11 R2 7/12 R1 5/11 B2 7/11 R2 5/12 B1 4/11 B2McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  34. 34. 5- 34 Bayes’ Theorem  Bayes’ Theorem is a method for revising a probability given additional information.  It is computed using the following formula: P( A1 ) P( B / A1 ) P( A1 | B) = P( A1 ) P( B / A1 ) + P( A2 ) P( B / A2 )McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  35. 35. 5- 35 EXAMPLE 9 Duff Cola Company recently received several complaints that their bottles are under-filled. A complaint was received today but the production manager is unable to identify which of the two Springfield plants (A or B) filled this bottle. What is the probability that the under-filled bottle came from plant A?McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  36. 36. 5- 36 EXAMPLE 9 continued The following table summarizes the Duff production experience. % of Total % of under- Production filled bottles A 55 3 B 45 4McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  37. 37. 5- 37 Example 9 continued P ( A) P (U / A) P( A / U ) = P ( A) P (U / A) + P ( B ) P (U / B ) .55(.03) = = .4783 .55(.03) +.45(.04)The likelihood the bottle was filled in Plant Ais reduced from .55 to .4783.McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  38. 38. 5- 38 Some Principles of Counting The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both. Example 10: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have? (10)(8) = 80McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  39. 39. 5- 39 Some Principles of Counting A permutation is any arrangement of r objects selected from n possible objects.  Note: The order of arrangement is important in permutations. n! n Pr = (n − r )!McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  40. 40. 5- 40 Some Principles of Counting A combination is the number of ways to choose r objects from a group of n objects without regard to order. n! nCr = r! (n − r )!McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  41. 41. 5- 41 EXAMPLE 11 There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick five players among the twelve on the team to comprise the starting lineup. How many different groups are possible? 12! 12C 5 = = 792 5! (12 − 5)!McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
  42. 42. 5- 42 Example 11 continued Suppose that in addition to selecting the group, he must also rank each of the players in that starting lineup according to their ability. 12! 12 P 5 = = 95,040 (12 − 5)!McGraw-Hill/Irwin Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.

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