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AP Statistics Monday , 11 November 2013

AP Statistics Monday , 11 November 2013. OBJECTIVE TSW investigate the basics of probability. ASSIGNMENTS DUE TOMORROW WS Counting Principle #1 WS Counting Principle #2 TOMORROW: QUIZ Counting Principle (includes permutations and combinations) Calendars will be given tomorrow.

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AP Statistics Monday , 11 November 2013

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  1. APStatisticsMonday, 11 November 2013 • OBJECTIVETSW investigate the basics of probability. • ASSIGNMENTS DUE TOMORROW • WS Counting Principle #1 • WS Counting Principle #2 • TOMORROW:QUIZ Counting Principle (includes permutations and combinations) • Calendars will be given tomorrow. • Everyone needs to have a calculator on their desk ready to use in class today.

  2. Probability I

  3. Sample space the collection of all possible outcomes of a chance experiment Roll a die S={1,2,3,4,5,6}

  4. Event • any collection of outcomes from the sample space • Rolling a prime # E= {2,3,5} • Rolling a prime # or even number E={2,3,4,5,6}

  5. Consists of all outcomes that are not in the event Not rolling an even # EC={1,3,5} Complement

  6. Union • the event A or B happening • consists of all outcomes that are in at least one of the two events • Rolling a prime # or even number E={2,3,4,5,6}

  7. Intersection • the event A and B happening • consists of all outcomes that are in both events • Drawing a red card and a “2” E = {2 hearts, 2 diamonds}

  8. two events have no outcomes in common Roll a “2” and a “5” Mutually Exclusive (disjoint)

  9. Venn Diagrams • Used to display relationships between events • Helpful in calculating probabilities

  10. Venn diagram Complement of A AC A

  11. Venn diagram A or B A B

  12. Venn diagram A and B A B

  13. Venn diagram Disjoint Events A B

  14. What does this show? Stat Cal Com Sci Com Sci Statistics & Computer Science & not Calculus

  15. What does this show? Cal Stat Com Sci Calculus or Computer Science

  16. What does this show? Stat Cal Com Sci (Statistics or Computer Science) and not Calculus

  17. What does this show? Cal Stat Com Sci Statistics and not (Computer Science or Calculus)

  18. How many are in -- Stat Cal 105 30 20 10 10 20 80 Com Sci Statistics or Computer Science? 170

  19. How many are in -- Stat Cal 105 30 20 10 10 20 80 Com Sci 20 Statistics and Computer Science?

  20. How many are in -- Stat Cal 105 30 20 10 10 20 80 Com Sci Statistics or (Computer Science and Calculus)? 90

  21. How many are in -- Stat Cal 105 30 20 10 10 20 80 Com Sci (Statistics or Computer Science) and Calculus? 50

  22. Assignment • WS Counting Principle #2 • Due tomorrow, 12 November 2013 • WS Probability • Due Thursday, 14 November 2013

  23. APStatisticsTuesday, 12 November 2013 • OBJECTIVETSW (1) investigate the basics of probability, and (2) quiz over the counting principle, permutations, and combinations. • ASSIGNMENTS DUE • WS Counting Principle #1  wire basket • WS Counting Principle #2  black tray • QUIZ Counting Principle (includes permutations and combinations) will be after the lesson.

  24. WSCounting Principles #1 • 6 categories • 5040 different arrangements • 390,700,800 different routes • 362,880 ways to arrange • 8,000,000 7-digit telephone #’s • 593,775 different teams • 1,000,000,000 different SSN • 725,760 arrangements • 43, 680 different slates • 9,765, 625 different answer keys

  25. WSCounting Principles #2 • 8.159152832 x 1047 different routes • 3,268,760 different combinations • Science: 86,450 different teamsMath: 1,218,000 different teams • 95,040 different arrangements • 36 different configurations • 2730 different ways • 27,405 different winning combinations • 105 three-digit numbers

  26. Probability II

  27. Denoted by P(Event) Probability This method for calculating probabilities is only appropriate when the outcomes of the sample space are equally likely.

  28. The relative frequency at which a chance experiment occurs Flip a fair coin 30 times & get 17 heads Experimental Probability

  29. As the number of repetitions of a chance experiment increases, the difference between the relative frequency of occurrence for an event and the true probability approaches zero. Law of Large Numbers

  30. Rule 1.Legitimate Values For any event E, 0 < P(E) < 1 Rule 2.Sample space If S is the sample space, P(S) = 1 Basic Rules of Probability

  31. Rule 3.Complement • For any event E, • P(E) + P(not E) = 1

  32. Rule 4.Addition • If two events E & F are disjoint, • P(E or F) = P(E) + P(F) • In general, • P(E or F) = P(E) + P(F) – P(E & F)

  33. APStatisticsThursday, 14 November 2013 • OBJECTIVETSWinvestigate the basics of probability and conditional probability. • EVERYONE needs a calculator. • TODAY’S ASSIGNMENT • WSConditional Probability

  34. Ex 1) A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. (Note: these are not simple events since there are many types of each brand.) Suppose that P(H) = 0.25, P(N) = 0.18, P(T) = 0.14. Are these disjoint events? yes P(H or N or T) = 0.25 + 0.18+ 0.14 = 0.57 P(not (H or N or T) = 1  0.57 = 0.43

  35. Ex. 2) Musical styles other than rock and pop are becoming more popular. A survey of college students finds that the probability they like country music is 0.40. The probability that they liked jazz is 0.30 and that they liked both is 0.10. What is the probability that they like country or jazz? P(C or J) = 0.4 + 0.3  0.1 = 0.6

  36. Two events are independent if knowing that one will occur (or has occurred) does not change the probability that the other occurs A randomly selected student is female - What is the probability she plays soccer for JVHS? A randomly selected student is female - What is the probability she plays football for JVHS? Independent Independent Not independent

  37. Rule 5.Multiplication If two events A & B are independent, General rule:

  38. Ex. 3) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that both bulbs are defective? Can you assume they are independent? Yes

  39. Ex 4) If P(A) = 0.45, P(B) = 0.35, and A & B are independent, findP(A or B). Are A & B disjoint? NO, independent events cannot be disjoint If A & B are disjoint, are they independent? Disjoint events do not happen at the same time. So, if A occurs, can B occur? Disjoint events are dependent! P(A or B) = P(A) + P(B) – P(A & B) If independent, multiply How can you find the probability of A & B? P(A or B) = 0.45 + 0.35 − 0.45(0.35) = 0.6425

  40. Ex 5) Suppose I will pick two cards from a standard deck without replacement. What is the probability that I select two spades? Are the cards independent? NO P(A & B) = P(A) · P(B|A) Read “probability of B given that A occurs” P(Spade & Spade) = 1/4 · 12/51 = 1/17 The probability of getting a spade given that a spade has already been drawn.

  41. Rule 6.At least one The probability that at least one outcome happens is 1 minus the probability that no outcomes happen. P(at least 1) = 1 – P(none)

  42. Ex. 6) A certain brand of light bulbs are defective five percent of the time. You randomly pick a package of two such bulbs off the shelf of a store. What is the probability that at least one bulb is defective? P(at least one) = P(D & DC) or P(DC & D) or P(D & D) = 1 - P(DC & DC) = 0.0975

  43. Dr. Pepper Ex 7) For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles. You buy a six-pack. What is the probability that you win something? P(at least one winning symbol) = 1 – P(no winning symbols) 1  0.96 = 0.4686

  44. A probability that takes into account a given condition Rule 7: Conditional Probability

  45. Ç 0.1 P(S F) = = = P(S | F) 0.1961 P(F) 0.51 Ex 8) In a recent study it was found that the probability that a randomly selected student is a girl is 0.51 and is a girl and plays sports is 0.10. If the student is female, what is the probability that she plays sports?

  46. Ç P(S M) x = = P(S | M) 0.31 P(M) 0.49 = x 0.1519 Ex 9) The probability that a randomly selected student plays sports if they are male is 0.31. What is the probability that the student is male and plays sports if the probability that they are male is 0.49?

  47. APStatisticsFriday, 15 November 2013 • OBJECTIVETSW work on WS’s of probability. • EVERYONE needs a calculator. • Copy the chart and questions (on the board) in your notes. • ASSIGNMENTS DUE MONDAY • WS Probability • WS Conditional Probability • Looking Ahead • Monday, 18 November 2013: QUIZ: Conditional Probability • Tuesday, 19 November 2013: REVIEW: Chapter 6 • Thursday, 21 November 2013: TEST: Chapter 6

  48. Conditional Probability Example 1: Only 5% of male high school basketball, baseball, and football players go on to play at the college level. Of these, only 1.7% enters major league professional sports. Of the athletes that do not play college sports, only 0.1% enters major league professional sports. What is the probability that a high school athlete will play professional sports? What is the probability that a high school athlete did not play college sports if he plays professional sports?

  49. pro (0.017) ~pro (0.983) college (0.05) ~college (0.95) pro (0.001) ~pro (0.999) 0.05(0.017) = 0.00085 0.05(0.983) = 0.04915 0.95(0.001) = 0.00095 0.95(0.999) = 0.94905 Σ = 1 What is the probability that a high school athlete will play professional sports? P(pro) = 0.00085 + 0.00095 = 0.0018 What is the probability that a high school athlete did not play college sports if he plays professional sports?

  50. Example 2: Management has determined that customers return 12% of the items assembled by inexperienced employees, whereas only 3% of the items assembled by experienced employees are returned. Due to turnover and absenteeism at an assembly plant, inexperienced employees assemble 20% of the items. Construct a tree diagram or a chart for this data. What is the probability that an item is returned? If an item is returned, what is the probability that an inexperienced employee assembled it? 0.048 0.5

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