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Probability

Probability. Revision 4. Question 1. An urn has two white balls and two black balls in it. Two balls are drawn out without replacing the first ball. a. What is the probability that the second ball is white, given that the first ball is white?

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Probability

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  1. Probability Revision 4

  2. Question 1 An urn has two white balls and two black balls in it. Two balls are drawn out without replacing the first ball. a. What is the probability that the second ball is white, given that the first ball is white? b. What is the probability that the first ball was white, given that the second ball is white?

  3. Question 1 An urn has two white balls and two black balls in it. Two balls are drawn out without replacing the first ball. • What is the probability that the second ball is white, given that the first ball is white? • If the first ball is white, there are now 3 balls left and only one is white: 1/3

  4. Question 1 An urn has two white balls and two black balls in it. Two balls are drawn out without replacing the first ball. b. What is the probability that the first ball was white, given that the second ball is white?

  5. Question 2

  6. Draw Venn diagrams

  7. P(A)=0.4, P(B)=0.6, P(A B)=0.3, Find 0.1 0.3 0.3 0.3

  8. P(A)=0.3, P(B)=0.4,Find 0.1 0.2 0.2 0.5

  9. 2c. Mutually Exclusive 0.3 0.6

  10. P(A)=0.5, find the probabilities of the events

  11. P(A)=0.5, Draw a Venn diagram 0.3 0.3 0.2

  12. d. P(A)=0.5, Draw a Venn diagram A B 0.3 0.3 0.2

  13. e. P(A)=0.4, Draw a Venn diagram A B 0.12 0.18 0.28 0.42

  14. Question 3 What does it mean for two events A and B to be statistically independent?

  15. Question 3 What does it mean for two events A and B to be statistically independent? It means that the occurrence of one does not affect the probability of the other.

  16. Question 4 On a certain type of aircraft the warning lights (showing green for normal and red for trouble) for the engines are accurate 90% of the time. If there are problems with the engines on 2% of all flights, find the probability that there is a fault with an engine, given that the warning light shows red.

  17. R 0.9 T 0.02 0.1 G R 0.1 0.98 T’ 0.9 G

  18. We can also create a table: Assume we look at 1000 flights “problems with the engines on 2% of flights”

  19. We can also create a table: Assume we look at 1000 flights “warning lights are accurate 90% of the time”

  20. We can also create a table: Assume we look at 1000 flights Finish the table

  21. find the probability that there is a fault with an engine, given that the warning light shows red.

  22. Question 5 One of the biggest problems with conducting a mail survey is the poor response rate. In an effort to reduce nonresponse, several different techniques for formatting questionnaires have been proposed. An experiment was conducted to study the effect of the questionnaire layout and page size on response in a mail survey. The results are given below.

  23. a. What proportion of the sample responded to the questionnaire?

  24. a. What proportion of the sample responded to the questionnaire?

  25. b. What proportion of the sample received the typeset small-page version?

  26. c. What proportion of those who received a typeset large-page version actually responded to the questionnaire?

  27. d. What proportion of the sample received a typeset large-page questionnaire and responded?

  28. e. What proportion of those who responded to the questionnaire actually received a type-written large page questionnaire?

  29. f. By looking at the response rates for each of the four formats, what do you conclude from the study?

  30. f. By looking at the response rates for each of the four formats, what do you conclude from the study? Type set (Large page) gave the best response rate at 68% with typewritten (large page) almost the same at 66% and Type set (Small page) was the worst at 51%. As a margin of error is likely I would conclude that the response rates seem to be better for large page.

  31. Question 6 A car park contains five Japanese cars and six non Japanese cars. A random variable X is defined by the number of Japanese cars among the first three cars to leave. a. Find the probability distribution of X. b. Calculate the expected number of Japanese cars to leave (among the first three cars to leave). c. Calculate the standard deviation.

  32. Question 6 A car park contains five Japanese cars and six non Japanese cars. A random variable X is defined by the number of Japanese cars among the first three cars to leave. • Find the probability distribution of X.

  33. Distribution Table

  34. Question 6 A car park contains five Japanese cars and six non Japanese cars. A random variable X is defined by the number of Japanese cars among the first three cars to leave. • Calculate the expected number of Japanese cars to leave (among the first three cars to leave).

  35. Distribution Table

  36. Expected number

  37. Variance

  38. Question 7 The mean salary at Carter-Drumfield sportswear store is $24 000 per annum with a standard deviation of $600. All employees get a $500 rise per annum. a. What will be the new mean salary and standard deviation of salaries? b. If instead of a flat rise, each employee had an income increase of 2%, what would be the new mean salary and standard deviation?

  39. Question 7 The mean salary at Carter-Drumfield sportswear store is $24 000 per annum with a standard deviation of $600. All employees get a $500 rise per annum. • What will be the new mean salary and standard deviation of salaries? Mean = 24 000 + 500 = $24 500 Standard deviation = $600

  40. Question 7 The mean salary at Carter-Drumfield sportswear store is $24 000 per annum with a standard deviation of $600. All employees get a $500 rise per annum. b. If instead of a flat rise, each employee had an income increase of 2%, what would be the new mean salary and standard deviation?

  41. Question 7 The mean salary at Carter-Drumfield sportswear store is $24 000 per annum with a standard deviation of $600. All employees get a $500 rise per annum. b. If instead of a flat rise, each employee had an income increase of 2%, what would be the new mean salary and standard deviation?

  42. Question 8 Adult males of a certain species are known to have a mean weight of 1.3 kg with a standard deviation of 0.2 kg. Adult females have a mean weight of 0.9 kg with a standard deviation of 0.1 kg. a. What is the mean weight and standard deviation of randomly selected groups of 2 males and 4 females? b. What is the probability that such a group will have a weight exceeding 6.5 kg? State your assumptions.

  43. Question 8 Adult males of a certain species are known to have a mean weight of 1.3 kg with a standard deviation of 0.2 kg. Adult females have a mean weight of 0.9 kg with a standard deviation of 0.1 kg. • What is the mean weight and standard deviation of randomly selected groups of 2 males and 4 females?

  44. Question 8 Adult males of a certain species are known to have a mean weight of 1.3 kg with a standard deviation of 0.2 kg. Adult females have a mean weight of 0.9 kg with a standard deviation of 0.1 kg. • What is the mean weight and standard deviation of randomly selected groups of 2 males and 4 females?

  45. Question 8 Adult males of a certain species are known to have a mean weight of 1.3 kg with a standard deviation of 0.2 kg. Adult females have a mean weight of 0.9 kg with a standard deviation of 0.1 kg. b. What is the probability that such a group will have a weight exceeding 6.5 kg? State your assumptions.

  46. Question 8 Adult males of a certain species are known to have a mean weight of 1.3 kg with a standard deviation of 0.2 kg. Adult females have a mean weight of 0.9 kg with a standard deviation of 0.1 kg. • What is the probability that such a group will have a weight exceeding 6.5 kg? State your assumptions. We assume that the weights are normally distributed. Mean = 6.2 SD = 0.346

  47. Question 8 • What is the probability that such a group will have a weight exceeding 6.5 kg? State your assumptions. We assume that the weights are normally distributed. Mean = 6.2 SD = 0.346 6.2 6.5

  48. Question 9 Two bags have black and white counters. Bag 1: 3 black and 1 white Bag 2: 6 black and 2 white. a. Which bag gives a better chance of picking a black counter? b. Which bag gives a better chance of picking two black counters?

  49. Question 9 Two bags have black and white counters. Bag 1: 3 black and 1 white Bag 2: 6 black and 2 white. • Which bag gives a better chance of picking a black counter? There is an equal chance

  50. Question 9 Two bags have black and white counters. Bag 1: 3 black and 1 white Bag 2: 6 black and 2 white. • Which bag gives a better chance of picking two black counters?

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