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Chapter 4

Chapter 4. Probabilities and Proportions. Chances of winning Lotto. Chances of winning Lotto. Which one has the higher chance of winning? First B. Second C. Neither (same chance). Roulette.

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Chapter 4

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  1. Chapter 4 Probabilities and Proportions

  2. Chances of winning Lotto

  3. Chances of winning Lotto Which one has the higher chance of winning? • First B. Second C. Neither (same chance)

  4. Roulette In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win.

  5. In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win. Roulette YES or NO?

  6. Let’s Make a Deal Game Show

  7. What is the better strategy? • Switch • Stay • It makes no difference Let’s Make a Deal Game Show

  8. Chapter 4 Probabilities and Proportions

  9. What are probabilities? A probability is a number between0 and 1 that quantifies uncertainty. The probability that an event A occurs is written as pr(A). 0 Impossible 1 Certain

  10. Examples: I toss a fair coin (where ‘fair’ means ‘equally likely outcomes’) • What are the possible outcomes? • What is the probability it will turn up heads? I choose a person at random and check which eye she/he winks with • What are the possible outcomes? H & T 1/2 L & R

  11. I toss a fair coin (where ‘fair’ means ‘equally likely outcomes’) What are the possible outcomes? What is the probability it will turn up heads? I choose a person at random and check which eye she/he winks with What are the possible outcomes? Examples: What is the probability they wink with their left eye? • One-half • One-quarter • Can’t tell H & T 1/2 L & R

  12. I toss a fair coin (where ‘fair’ means ‘equally likely outcomes’) What are the possible outcomes? What is the probability it will turn up heads? I choose a person at random and check which eye she/he winks with What are the possible outcomes? What is the probability they wink with their left eye? Examples: H & T 1/2 L & R ?

  13. pr(A ) = Equally likely outcomes For equally likely outcomes: number of outcomes in A total number of outcomes The probability of getting a four when a fair dice is rolled is 1/6

  14. Probabilities and proportions Probabilities and proportions are numerically equivalent. • The proportion of New Zealanders who are left handed is 0.1. • A randomly selected New Zealander is left handed with a probability of 0.1.

  15. House Sales Let A be the event that a sale is made within 3 weeks B be the event that a sale is over $600,000

  16. House Sales (a) What proportion of these sales were over $600,000? pr(B ) = 129/343 = 0.38

  17. House Sales (b) What proportion of these sales were not over $600,000? pr(B) = (28+186)/343 = 0.62

  18. House Sales (c) What proportion of these sales were made in 3 or more weeks? pr(A) = (121+88)/343 = 0.61

  19. House Sales (d) What proportion of these sales were made within 3 weeks and sold for over $600,000?

  20. House Sales (d) • pr(A) B. pr(A and B) C. pr(B) D. pr(A or B) E. I don’t know What proportion of these sales were made within 3 weeks and sold for over $600,000?

  21. House Sales (d) • 52/134 B. 52/343 C. 52/129 D. 211/343 E. I don’t know What proportion of these sales were made within 3 weeks and sold for over $600,000?

  22. House Sales (d) What proportion of these sales were made within 3 weeks and sold for over $600,000? pr(A and B ) = 52/343 = 0.15

  23. House Sales (e) What proportion of these sales were made within 3 weeks or sold for over $600,000? pr(A or B ) = (134+129-52)/343 = 0.62

  24. House Sales (f) What proportion of these sales were on the market for less than 3 weeks given that they sold for over $600,000?

  25. House Sales (f) What proportion of these sales were on the market for less than 3 weeks given that they sold for over $600,000? 52/129 = 0.40

  26. Conditional Probabilities The sample space is reduced. Key words that indicate conditional probability are: given that, of those, if, assuming that “The probability of event A occurring given that event B has already occurred” is written in shorthand as: pr(A|B)

  27. House Sales (g) What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000?

  28. House Sales (g) The event in this question is? A. Single B. Joint C. Conditional D. I don’t know What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000?

  29. House Sales (g) Conditional probability? A. Yes B. No C. I don’t know What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000?

  30. House Sales (g) What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000? pr(B|A) = 52/134 = 0.39

  31. Filled jobs by industry and type (a) Working owner What proportion of workers were part time employees?

  32. Filled jobs by industry and type (a) The event in this question is? A. Single B. Joint C. Conditional D. I don’t know Working owner What proportion of workers were part time employees?

  33. Filled jobs by industry and type (a) Conditional probability? A. Yes B. No C. I don’t know Working owner What proportion of workers were part time employees?

  34. Filled jobs by industry and type (a) Working owner What proportion of workers were part time employees? pr(PT) = 458/1646 = 0.28

  35. Filled jobs by industry and type (b) Working owner The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion?

  36. Filled jobs by industry and type (b) The event in this question is? A. Single B. Joint C. Conditional D. I don’t know Working owner The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion?

  37. Filled jobs by industry and type (b) Conditional probability? A. Yes B. No C. I don’t know Working owner The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion?

  38. Filled jobs by industry and type (b) Working owner The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? pr(PT |A) = 57/99 = 0.58

  39. Filled jobs by industry and type (c) What proportion of workers were in the retail trade?

  40. Filled jobs by industry and type (c) The event in this question is? A. Single B. Joint C. Conditional D. I don’t know What proportion of workers were in the retail trade?

  41. Filled jobs by industry and type (c) Conditional probability? A. Yes B. No C. I don’t know What proportion of workers were in the retail trade?

  42. Filled jobs by industry and type (c) What proportion of workers were in the retail trade? pr(R) = 232/1646 = 0.14

  43. Filled jobs by industry and type (d) What proportion of workers were full time employees working in education?

  44. Filled jobs by industry and type (d) The event in this question is? A. Single B. Joint C. Conditional D. I don’t know What proportion of workers were full time employees working in education?

  45. Filled jobs by industry and type (d) Conditional probability? A. Yes B. No C. I don’t know What proportion of workers were full time employees working in education?

  46. Filled jobs by industry and type (d) What proportion of workers were full time employees working in education? pr(FT and E) = 87/1646 = 0.05

  47. Response Rates by Survey Format (a) What proportion of the students received an incentive and responded? 125/4416 = 0.03

  48. Response Rates by Survey Format (b) What was the overall response rate to the survey? 948/4416 = 0.21

  49. Response Rates by Survey Format (c) Which format had the highest response rate? Try it!!!!

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