ODD ANSWERS A61 3. (a) Without the requirement, the number of possible PINs is 105 = 100,000. With the requirement, the number of possible PINs is 10P5 = 10# 9# 8# 7# 6 = 30,240. (b) Sample answer: No, although the requirement would likely discourage customers from choosing predictable PINs, the numbers of possible PINs would significantly decrease, and the most popular PIN, 12345, would still be allowed. Chapter 4 Section 4.1 (page 197) 1. A random variable represents a value associated with each outcome of a probability experiment. Examples: Answers will vary. 3. No. The expected value may not be a possible value of x for one trial, but it represents the average value of x over a large number of trials. 5. False. In most applications, discrete random variables represent counted data, while continuous random variables represent measured data. 7. False. The mean of the random variable of a probability distribution describes a typical outcome. The variance and standard deviation of the random variable of a probability distribution describe how the outcomes vary. 9. Discrete. Attendance is a random variable that is countable. 11. Continuous. Distance traveled is a random variable that must be measured. 13. Discrete. The number of cars in a university parking lot is a random variable that is countable. 15. Continuous. The volume of blood drawn for a blood test is a random variable that must be measured. 17. Discrete. The fitted sizes of hats is a random variable that can be measured. 19. (a) x P(x) 0 0.028 1 0.256 2 0.340 3 or more 0.376 (b) Number of televisions Probability 0 1 3 or more 2 Televisions P(x) 0.10 0.20 0.30 0.40 0.05 0.15 0.25 0.35 x Skewed left 21. (a) 0.596 (b) 0.716 (c) 0.972 (d) 0.624 23. Yes, it would be unusual for a household to have no HD televisions because the probability is less than or equal to 0.05. 25. 0.34 27. Yes 29. (a) m ≈ 0.5; s 2 ≈ 0.8; s ≈ 0.9 (b) The mean is 0.5, so the average number of dogs per household is about 0 or 1 dog. The standard deviation is 0.9, so most of the households differ from the mean by no more than about 1 dog. 31. (a) m ≈ 1.5; s 2 ≈ 1.5; s ≈ 1.2 (b) The mean is 1.5, so the average batch of 1000 machine parts has 1 or 2 defects. The standard deviation is 1.2, so most of the batches of 1000 differ from the mean by no more than about 1 defect. 33. (a) mean = 2.01 variance = 1.07 standard deviation = 1.03 (b) The mean is 2.01, so the average category of hurricane is about 2. The standard deviation is 1.03, so most hurricanes differ from the mean by no more than about 1 category. 35. An expected value of 0 means that the money gained is equal to the money spent, representing the break-even point. 37. -$0.05 39. Area of bar 1 = 0.028, area of bar 2 = 0.256, area of bar 3 = 0.340, area of bar 4 = 0.376, sum = 1. The area of each of the bars is the probability of each outcome because the width is one. The sum is 1 is because all of the probabilities should add up to 1, as they are between 0 and 1. 41. $47,980 43. mean = 1047; standard deviation ≈ 164.9 Section 4.2 (page 210) 1. Each trial is independent of the other trials when the outcome of one trial does not affect the outcome of any of the other trials. 3. c. Because the probability is greater than 0.5, the distribution is skewed left. 4. b. Because the probability is 0.5, the distribution is symmetric. 5. a. Because the probability is less than 0.5, the distribution is skewed right. 6. c. The histogram shows probabilities for 12 trials. 7. a. The histogram shows probabilities for 4 trials. 8. b. The histogram shows probabilities for 8 trials. As n increases, the distribution becomes more symmetric. 9. Combinations Rule 11. m = 20; s 2 = 12; s ≈ 3.5 13. m ≈ 32.2; s 2 ≈ 23.9; s ≈ 4.9 15. Binomial experiment, Success: frequent gamers who own a virtual reality device n = 10; p = 0.29; q = 0.71; x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 17. Not a binomial experiment because the probability of a success is not the same for each trial 19. (a) 0.037 (b) 0.281 21. (a) 0.188 (b) 0.207 (c) 0.329 23. (a) 0.341 (b) 0.998 (c) 0.056 25. (a) 0.123 (b) 0.030 (c) 0.153
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