1. Welcome to MM150 – Unit 7 Seminar Instructor: Larry Musolino Email: lmusolino@kaplan.edulmusolino@kaplan.edu 7.1 – Nature of Probability 7.2 – Theoretical Probability 7.3 – Odds 7.4 – Expected Value 7.5 – OR and AND Problems

2. Reminder on Final Project START THINKING ABOUT TOPIC FOR FINAL PROJECT! Final Project is due at the end of Unit 9. You will post your Final Project to Dropbox in Unit 9. For more details on Final Project, see Unit 6 topic “Final Project” – You will select a topic in the course and discuss a potential application for this concept in your chosen profession. Do not be afraid to "think outside the box" when discussing an application in your profession. This also could be an example of how the concept learned is fundamental to understanding a more complex concept.

3. Reminder on Final Project (cont’d) Final Project Instructions: Choose either Microsoft Word or Microsoft PowerPoint in which to create your project. This course does not teach PowerPoint, so if you choose this format, do so only because you are already comfortable with it or know someone who can help you learn to use it. Create 5 slides or pages. On slides or pages 1 and 2, provide your name, the project title, and the course and section number, and introduce your chosen profession and give a brief overview of the concept you will apply to the profession. On slides or pages 3 and 4, describe how the concept can apply to you chosen profession. You will not need to “do the math”; simply describe how you would use it and provide examples of situations in which you would use the concept you have chosen. On slide or page 5, provide any resources you have used to give credit to others’ ideas and information. Check spelling and grammar and visit the Writing Center if needed. Submit your final project to the Unit 9 dropbox for grading. You will have an opportunity to share with your classmates at the Math Fair in Unit 10

4. Reminder on Final Project (cont’d) More Notes: See the example Final Projects Posted under: – Unit 6 Heading or Unit 7 Heading – Then Click on Icon for Final Project in Main Window Under Unit 8, heading “Final Project Introduction” – This is Discussion Board for you to post your preliminary ideas for a final project, classmates and instructor can then offer comments/suggestions/etc.

5. 7.1 The Nature of Probability

6. Definitions An experiment is a controlled operation that yields a set of results. – Example: Roll a die. – Example: Record the next 20 birds at the bird feeder. The possible results of an experiment are called its outcomes. – Example: When rolling a die, the outcome is either 1, 2, 3, 4, 5, 6. An event is a subcollection of the outcomes of an experiment. – Example: Rolling a die and an even number results.

7. Empirical and Theoretical Probability Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment. Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment.

8. Empirical Probability Example: In 100 tosses of a fair die, 19 landed showing a 3. Find the empirical probability of the die landing showing a 3. Let E be the event of the die landing showing a 3.

9. The Law of Large Numbers The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals. – If you rolled a die many times, (say 100,000 times) the law of large numbers states that the probability of rolling a 3 will be 1/6 or 0.167 or about 16%. – The larger the number of trials, the closer the observed probability will tend towards the theoretical probability.

10. You Try It #1 The last 30 birds that fed at the bird feeder in your backyard were 14 finches, 10 cardinals and 6 blue jays. Find the empirical probability that the next bird to feed from the feeder is: – (a) Finch – (b) Cardinal – (c) Blue Jay

11. You Try It #1 - Solution The last 30 birds that fed at the bird feeder in your backyard were 14 finches, 10 cardinals and 6 blue jays. Find the empirical probability that the next bird to feed from the feeder is: – (a) FinchP(Finch) = 14/30 = 0.47 = 47% – (b) CardinalP(Cardinal) = 10/30 = 0.33 = 33% – (c) Blue JayP(Blue Jay) = 6/30 = 0.20 = 20%

12. You Try It #2 The theoretical probability of rolling a 4 on a die is 1/6. Does this probability mean that if a die is rolled six times, then one “4” will be rolled? – If not, what does the statement “The theoretical probability of rolling a 4 on a die is 1/6” mean?

13. You Try It #2 - Solution The theoretical probability of rolling a 4 on a die is 1/6. Does this probability mean that if a die is rolled six times, then one “4” will be rolled? No If not, what does the statement “The theoretical probability of rolling a 4 on a die is 1/6” mean? – Over the long run, on average, one out of every six rolls will result in a “4”.

14. You Try It #3 In a random sample of 100 people, adults were asked their occupation. The results are per the table below: Find the probability that an adult selected at random will be in the education field. Health Care Field34 Accounting Field12 Education Field22 Engineering Field8 Transporatation Field18 Other6

15. You Try It #3 - Solution In a random sample of 100 people, adults were asked their occupation. The results are per the table below: Find the probability that an adult selected at random will be in the education field. – Probability = 22/100 = 0.22 = 22% Health Care Field34 Accounting Field12 Education Field22 Engineering Field8 Transporatation Field18 Other6

16. 7.2 Theoretical Probability

17. Equally likely outcomes If each outcome of an experiment has the same chance of occurring as any other outcome, they are said to be equally likely outcomes. For equally likely outcomes, the probability of Event E may be calculated with the following formula.

18. Example A die is rolled. Find the probability of rolling a) a 2. b) an odd number. c) a number less than 4. d) an 8. e) a number less than 9.

19. Solutions: There are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. a) b) There are three ways an odd number can occur 1, 3 or 5. c) Three numbers are less than 4.

20. Solutions: There are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. continued d) There are no outcomes that will result in an 8. e) All outcomes are less than 9. The event must occur and the probability is 1.

21. Important Facts The probability of an event that cannot occur is 0. The probability of an event that must occur is 1. Every probability is a number between 0 and 1 inclusive; that is, 0 <= P(E) <= 1. – Thus there can be no such thing as: Negative probability Probability greater than 1 The sum of the probabilities of all possible outcomes of an experiment is 1.

22. Example A standard deck of cards is well shuffled, and one card is selected at random. Find the probability that the card selected is…. a) a 10. b) not a 10. c) a heart. d) an ace, 2 or 3. e) diamond and spade. f) a card greater than 4 and less than 7.

23. Example continued a) a 10 There are four 10’s in a deck of 52 cards. b) not a 10

24. Example continued c) a heart There are 13 hearts in the deck. d) an ace, 2 or 3 There are 4 aces, 4 twos and 4 threes, or a total of 12 cards.

25. Example continued d) diamond and spade The word and means both events must occur. This is not possible. e) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s, and 6’s.

26. You Try It #4 You are taking a multiple choice test and you guess at the answer. – If a question has five possible choices, what is the probability that you guess the answer correctly? – Lets say you are able to eliminate one answer from the five choices but you still guess at an answer, what is the probability that you guess the answer correctly?

27. You Try It #4 – Solution You are taking a multiple choice test and you guess at the answer. – If a question has five possible choices, what is the probability that you guess the answer correctly? Probability = 1/5 = 0.20 = 20% – Lets say you are able to eliminate one answer from the five choices but you still guess at an answer, what is the probability that you guess the answer correctly? Probability = 1/4 = 0.25 = 25%

28. You Try It #5 A picnic cooler contains 100 cans of soda covered by ice. In the cooler there are 30 cans of Coke, 40 cans of orange soda, 10 cans of ginger ale, and 20 cans of root beer. If one can is selected at random, find the probability that: (A) can of root beer is selected (B) can of root beer or orange soda is selected. (C) can of Coke, root beer or orange soda is selected.

29. You Try It #5 - Solution A picnic cooler contains 100 cans of soda covered by ice. In the cooler there are 30 cans of Coke, 40 cans of orange soda, 10 cans of ginger ale, and 20 cans of root beer. If one can is selected at random, find the probability that: (A) can of root beer is selected – Probability = 20/100 = 0.20 = 20% (B) can of root beer or orange soda is selected. – Probability = (20 + 40)/ 100 = 60/100 = 0.60 = 60% (C) can of Coke, root beer or orange soda is selected. – Probability = (30 + 20 + 40)/ 100 = 90/100 = 0.90 = 90%

30. You Try It #6 One card is selected at random from deck of cards. Find the probability that the card selected is: (A) a diamond (B) a red card (C) a “5” or a “7” (D) not a heart (E) a red card or a black card

31. You Try It #6 - Solution One card is selected at random from deck of cards. Find the probability that the card selected is: (A) a diamond – Probability = 13/52 = 0.25 = 25% (B) a red card – Probability = 26/52 = 0.50 = 50% (C) a “5” or a “7” – Probability = (4 + 4)/ 52 = 0.15 = 0.15% (D) not a heart – Probability = 26/52 = 0.50 = 50% (E) a red card or a black card – Probability = 52/52 = 1.0 = 100%

32. 7.3 Odds

33. Odds Against

34. Reminder on Fractions To multiply two fractions – multiply across on top and on bottom: To divide fractions – change the divison to multiplication and flip (invert) the 2 nd fraction: flip

35. Example: Odds Against Example: Find the odds against rolling a 5 on one roll of a die. The odds against rolling a 5 are 5:1. odds against rolling a 5 Note how the denominators will cancel out. Reminder: To divide two fractions, change to multiplication and flip the 2 nd fraction

36. Odds in Favor

37. Example Find the odds in favor of landing on blue in one spin of the spinner. The odds in favor of spinning blue are 3:5.

38. Probability from Odds Example: The odds against spinning a blue on a certain spinner are 4:3. Find the probability that a) a blue is spun. b) a blue is not spun.

39. Solution Since the odds are 4:3 the denominators must be 4 + 3 = 7. The probabilities ratios are: Note: If the odds against an event are a:b, then the odds in favor of the event are b:a

40. You Try It #7 A card is picked at random from deck of cards. Find the ODDS AGAINST and ODDS IN FAVOR for selecting: (A) a Queen (B) a Picture Card

41. You Try It #7 - Solution A card is picked at random from deck of cards. Find the ODDS AGAINST and ODDS IN FAVOR for selecting: (A) a Queen ODDS AGAINST = P(failure)/P(success) = (48/52) / (4/52) = (48/52)(52/4) = 48/4 = 12/1 = 12:1 ODDS IN FAVOR = P(success)/P(failure) = (4/52) / (48/52) = (4/52)(52/48) = 4/48 = 1/12 = 1:12 (B) a Picture Card ODDS AGAINST = P(failure)/P(success) = (40/52) / (12/52) = (40/52)(52/12) = 40/12 = 10/3 = 10:3 ODDS IN FAVOR = P(success)/P(failure) = (12/52) / (40/52) = (12/52)(52/40) = 12/40 = 3/10 = 3:10

42. 7.4 Expected Value (Expectation)

43. Expected Value The symbol P 1 represents the probability that the first event will occur, and A 1 represents the net amount won or lost if the first event occurs. Expected value is used to determine expected results of experiment or business venture over the long term.

44. Example Teresa is taking a multiple-choice test in which there are four possible answers for each question. The instructor indicated that she will be awarded 3 points for each correct answer and she will lose 1 point for each incorrect answer and no points will be awarded or subtracted for answers left blank. – If Teresa does not know the correct answer to a question, is it to her advantage or disadvantage to guess? – If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at the answer?

45. Solution Expected value if Teresa guesses. Gain is written as postive number, Loss is written as negative number Thus over long term she will neither gain nor lose points by guessing

46. Solution continued—eliminate a choice

47. You Try It #8 In the TV show, Deal or No Deal, a contestant is offered an amount of money based on dollar amounts contained in suitcases. Lets say a contestant has 4 suitcases left with following dollar amounts: $10, $500, $1000, $10,000. What is the expected value for the contestant?

48. You Try It #8 - Solution In the TV show, Deal or No Deal, a contestant is offered an amount of money based on dollar amounts contained in suitcases. Lets say a contestant has 4 suitcases left with following dollar amounts: $10, $500, $1000, $10,000. What is the expected value for the contestant? – The probability of selecting each dollar amount is ¼ or 0.25. – Thus the expected value is: Expected Value =0.25(10) + 0.25(500) + 0.25(1000)+0.25(10000) Expected Value = 2.5 + 125 + 250 + 2500 Expected Value = $2877.5

49. Example: Winning a Prize When Calvin Winters attends a tree farm event, he is given a free ticket for the $75 door prize. A total of 150 tickets will be given out. Determine his expectation of winning the door prize. Thus his expectation is ½ or 0.5 or $0.50

50. Example When Calvin Winters attends a tree farm event, he is given the opportunity to purchase a ticket for the $75 door prize. The cost of the ticket is $3, and 150 tickets will be sold. Determine Calvin’s expectation if he purchases one ticket.

51. Solution Calvin’s expectation is  $2.48 when he purchases one ticket.

52. Fair Price Fair price = expected value + cost to play Fair Price is the amount to be paid such that the expected value is zero.

53. Example Suppose you are playing a game in which you spin the pointer shown in the figure, and you are awarded the amount shown under the pointer. If is costs $10 to play the game, determine a) the expectation of the person who plays the game. b) the fair price to play the game. $10 $2 $20$15

54. Solution $0 3/8 $10 $5  $8 Amount Won/Lost 1/8 3/8Probability $20$15$2Amt. Shown on Wheel

55. Solution Fair price = expectation + cost to play =  $1.13 + $10 = $8.87 Thus, the fair price is about $8.87.

56. 7.5 Tree Diagrams

57. Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M N distinct ways.

58. Definitions Sample space: A list of all possible outcomes of an experiment. Sample point: Each individual outcome in the sample space. Tree diagrams are helpful in determining sample spaces.

59. Example Two balls are to be selected without replacement from a bag that contains one purple, one blue, and one green ball. a)Use the counting principle to determine the number of points in the sample space. b)Construct a tree diagram and list the sample space. c)Find the probability that one blue ball is selected. d)Find the probability that a purple ball followed by a green ball is selected.

60. Solutions a) 3 2 = 6 ways b) c) d) B P B G B G P G P PB PG BP BG GP GB

61. P(event happening at least once)

62. 7.6 Or and And Problems

63. Or Problems P(A or B) = P(A) + P(B)  P(A and B) Example: Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains an even number or a number greater than 5.

64. Solution P(A or B) = P(A) + P(B)  P(A and B) Thus, the probability of selecting an even number or a number greater than 5 is 7/10.

65. Example Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a bowl and one is randomly selected. Find the probability that the piece of paper selected contains a number less than 3 or a number greater than 7.

66. Solution There are no numbers that are both less than 3 and greater than 7. Therefore,

67. Mutually Exclusive Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously.

68. Example One card is selected from a standard deck of playing cards. Determine the probability of the following events. a) selecting a 3 or a jack b) selecting a jack or a heart c) selecting a picture card or a red card d) selecting a red card or a black card

69. Solutions a) 3 or a jack b) jack or a heart

70. Solutions continued c) picture card or red card d) red card or black card

71. And Problems P(A and B) = P(A) P(B) Example: Two cards are to be selected with replacement from a deck of cards. Find the probability that two red cards will be selected.

72. Example Two cards are to be selected without replacement from a deck of cards. Find the probability that two red cards will be selected.

73. Independent Events Event A and Event B are independent events if the occurrence of either event in no way affects the probability of the occurrence of the other event. Experiments done with replacement will result in independent events, and those done without replacement will result in dependent events.

74. Example A package of 30 tulip bulbs contains 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. Three bulbs are randomly selected and planted. Find the probability of each of the following. a.All three bulbs will produce pink flowers. b.The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower. c.None of the bulbs will produce a yellow flower. d.At least one will produce yellow flowers.

75. Solution 30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. a. All three bulbs will produce pink flowers.

76. Solution 30 tulip bulbs : 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. b. The first bulb selected will produce a red flower, the second will produce a yellow flower and the third will produce a red flower.

77. Solution 30 tulip bulbs, 14 bulbs for red flowers, 0010 for yellow flowers, and 6 for pink flowers. c. None of the bulbs will produce a yellow flower.

78. Solution 30 tulip bulbs, 14 bulbs for red flowers, 10 for yellow flowers, and 6 for pink flowers. d. At least one will produce yellow flowers. P(at least one yellow) = 1  P(no yellow)