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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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**Chapter 4**Probabilities and Proportions**Chances of winning Lotto**Which one has the higher chance of winning? • First B. Second C. Neither (same chance)**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.**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?**What is the better strategy?**• Switch • Stay • It makes no difference Let’s Make a Deal Game Show**Chapter 4**Probabilities and Proportions**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**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**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**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 ?**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**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.**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**House Sales (a)**What proportion of these sales were over $600,000? pr(B ) = 129/343 = 0.38**House Sales (b)**What proportion of these sales were not over $600,000? pr(B) = (28+186)/343 = 0.62**House Sales (c)**What proportion of these sales were made in 3 or more weeks? pr(A) = (121+88)/343 = 0.61**House Sales (d)**What proportion of these sales were made within 3 weeks and sold for over $600,000?**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?**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?**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**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**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?**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**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)**House Sales (g)**What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000?**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?**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?**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**Filled jobs by industry and type (a)**Working owner What proportion of workers were part time employees?**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?**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?**Filled jobs by industry and type (a)**Working owner What proportion of workers were part time employees? pr(PT) = 458/1646 = 0.28**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?**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?**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?**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**Filled jobs by industry and type (c)**What proportion of workers were in the retail trade?**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?**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?**Filled jobs by industry and type (c)**What proportion of workers were in the retail trade? pr(R) = 232/1646 = 0.14**Filled jobs by industry and type (d)**What proportion of workers were full time employees working in education?**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?**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?**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**Response Rates by Survey Format (a)**What proportion of the students received an incentive and responded? 125/4416 = 0.03**Response Rates by Survey Format (b)**What was the overall response rate to the survey? 948/4416 = 0.21**Response Rates by Survey Format (c)**Which format had the highest response rate? Try it!!!!

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