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Chances of winning Lotto

Learn about probabilities in games like Lotto, roulette, and Let's Make a Deal. Understand conditional probabilities and how to make strategic decisions. Improve your chances of winning!

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Chances of winning Lotto

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  1. Chances of winning Lotto

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

  3. 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.

  4. 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?

  5. Let’s Make a Deal Game Show

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

  7. Let’s Make a Deal Game Show If Door 1 is chosen: 2 3 3 2 1 2 or 3

  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 Abe the event that a sale is made in March • B be the event that a sale is at least $1 million

  16. House Sales (a) What proportion of these sales were at least $1 million? • pr(B ) = 954/19693 = 0.05

  17. House Sales (b) What proportion of these sales were less than $1million? • pr(B) = (10322+5158+3259)/19693 = 0.95

  18. House Sales (c) What proportion of these sales were made in January or February? • pr(A) = 1 – 8128/19693 = 0.59

  19. House Sales (d) What proportion of these sales were made in March and sold for at least $1million?

  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 in March and sold for at least $1million?

  21. House Sales (d) • 492/8128 B. 492/19693 C. 492/954 • D. 14760/19693 E. I don’t know What proportion of these sales were made in March and sold for at least $1million?

  22. House Sales (d) What proportion of these sales were made in March and sold for at least $1million? • pr(A and B ) = 492/19693 = 0.02

  23. House Sales (e) What proportion of these sales were made in March or sold for at least $1million? • pr(A or B ) = (8128+954-492)/19693 = 0.44

  24. House Sales (f) What proportion of these sales were made in March given that they sold for at least $1million?

  25. House Sales (f) What proportion of these sales were made in March given that they sold for at least $1million? • 492/954 = 0.52

  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 March, sold for at least $1million?

  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 March, sold for at least $1million?

  29. House Sales (g) Conditional probability? A. Yes B. No C. I don’t know What proportion of the houses that sold in March, sold for at least $1million?

  30. House Sales (g) What proportion of the houses that sold in March, sold for at least $1million? • pr(B|A) = 492/8128 = 0.06

  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) (d) What proportion of workers were full time employees working in education? Answer is: A. 87/1646 B. 87/125 C. 87/1056

  44. Answer is: A. 87/1646 B. 87/125 C. 87/1056 Filled jobs by industry and type (d) • (d) What proportion of workers were full time employees working in education?

  45. Answer is: A. 87/1646 B. 87/125 C. 87/1056 Filled jobs by industry and type (d) • (d) What proportion of workers were full time employees working in education? • What proportion of full time workers were working in education?

  46. Answer is: A. 87/1646 B. 87/125 C. 87/1056 Filled jobs by industry and type (d) • (d) What proportion of workers were full time employees working in education? • What proportion of full time workers were working in education?

  47. Answer is: A. 87/1646 B. 87/125 C. 87/1056 Filled jobs by industry and type (d) • (d) What proportion of workers were full time employees working in education? • What proportion of full time workers were working in education? • What proportion of workers in education were working full time?

  48. Answer is: A. 87/1646 B. 87/125 C. 87/1056 Filled jobs by industry and type (d) • (d) What proportion of workers were full time employees working in education? • What proportion of full time workers were working in education? • What proportion of workers in education were working full time?

  49. Answer is: A. 87/1646 B. 87/125 C. 87/1056 Filled jobs by industry and type (d) • (d) What proportion of workers were full time employees working in education? • What proportion of full time workers were working in education? • What proportion of workers in education were working full time? What proportion of workers were full time and were in education?

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