library(VennDiagram)
## Loading required package: grid
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Chapter 2
Problem 6,8,20,30,38,44
Problem 6
If you roll a pair of fair dice, the probability of
(a) getting a sum of 1 is 0 percent.
(b) getting a sum of 5 is 4/36.
(c) getting a sum of 12 is 1/36.
Problem 8
(a) Living below the poverty line and speaking a foreign language at home are not disjoint?
(b) Draw a Venn diagram summarizing the variables and their associated probabilities:
## (polygon[GRID.polygon.1], polygon[GRID.polygon.2], polygon[GRID.polygon.3], polygon[GRID.polygon.4], text[GRID.text.5], text[GRID.text.6], text[GRID.text.7], text[GRID.text.8], text[GRID.text.9])
(c) 10.4 percent of Americans live below the poverty line and only speak English at home.
(d) 31.1 percent of Americans live below the poverty line or speak a foreign language at home.
(e) 68.9 percent of Americans live above the poverty line and only speak English at home.
(f) The event that someone lives below the poverty line is not independent of the event that the person speaks a foreign language at home.
.146 * .207 = .0302, which does not equal .042
Problem 20
(a) What is the probability that a randomly chosen male respondent or his partner has blue eyes?
(114+108-78)/204
(b) What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes?
(78/114) = .684
(c) What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes?
(19/54) = .352
What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?
(11/36) = .306 (d) It doesn’t appear that the eye colors of male respondents and their partners are independent? blue male | blue female = (78/108) = .722
blue male | brown female(23/55) = .418
The probability of having blue eyes for a male is much higher if his partner has blue eyes.
Problem 30
(a) The probability of drawing a hardcover book ???rst then a paperback ???ction book second is:
(28/95)(67/94) = .21
(b) The probability of drawing a ???ction book ???rst and then a hardcover book second, without replacement is:
(72/95)((13/7227/94) + (59/7228/94)) = .2243
(c) Part (b), except with replacement:
(28/9572/95) = .2234 (d) The ???nal answers to parts (b) and (c) are very similar. Explain why:
The probability tends to go down as more choices are made. The higher the number of books, the less difference it will make when one is removed first.
Problem 38
(a) The average is .34(25) + .12(35) = $12.70
The variance is (.34(25^2) + .12(35^2)) - 12.7^2 = 198.21
The standard deviation is sqrt(198.21) = $14.08
(b) For 120 passengers, the expectation is (12.7)(120) = $1524
The standard deviation of the sum is sqrt(198.21 * 120) = $154.22
We assume they’re all from the same distribution. This is reasonable in this case.
Problem 44
(a) The distribution is a histogram based on an ogive, consisting of 9 categories and censored at $100,000
(b) .622 is the probability that a randomly chosen US resident makes less than $50,000 per year.
(c) The probability that a randomly chosen US resident makes less than$50,000 per year and is female is greater than .255. It would
be unreasonable to assume that the distribution is the same for woman as men and women together. This pay gap is well documented.
(d) This leads to an estimate of .718 * .41 = .294, which is indeed greater than .255. It was ok to assume that the
previous estimate was lower than the actual value.