Dice rolls. (3.6, p. 92) If you roll a pair of fair dice, what is the probability of

  1. getting a sum of 1?

0 as the lowest pair is a sum of 2.

  1. getting a sum of 5?

4 chances out of 36 total chances, 0.111

  1. getting a sum of 12?

1 chance out of 36 total chances, 0.0278

Poverty and language. (3.8, p. 93) The American Community Survey is an ongoing survey that provides data every year to give communities the current information they need to plan investments and services. The 2010 American Community Survey estimates that 14.6% of Americans live below the poverty line, 20.7% speak a language other than English (foreign language) at home, and 4.2% fall into both categories.

  1. Are living below the poverty line and speaking a foreign language at home disjoint?

No, these variables are not disjoint because some Americans can live below the poverty line and speak a language other than English, so the variables are not mutually exclusive.

  1. Draw a Venn diagram summarizing the variables and their associated probabilities.
library(VennDiagram)
## Loading required package: grid
## Loading required package: futile.logger
venn_plot <- draw.pairwise.venn(10.4, 16.3, 4.2, c("Americans living below the poverty line","Americans speaking a language other than English"), cat.pos = 0)

  1. What percent of Americans live below the poverty line and only speak English at home?

10.4% of Americans living below the poverty line only speak English at home.

  1. What percent of Americans live below the poverty line or speak a foreign language at home?

P(A) + P(B) - P(both) = 14.6% + 20.7% - 4.2% = 31.1%

  1. What percent of Americans live above the poverty line and only speak English at home?

1 - P(A or B) = 1 - 0.311 = 0.689 = 68.9%

  1. Is the event that someone lives below the poverty line independent of the event that the person speaks a foreign language at home?

If P(A)P(B) == P(A and B) then the events are independent. 0.146 0.207 = 0.030.

3% and 4.2 are not equal so the events are dependent.

Assortative mating. (3.18, p. 111) Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 204 Scandinavian men and their female partners. The table below summarizes the results. For simplicity, we only include heterosexual relationships in this exercise.

  1. What is the probability that a randomly chosen male respondent or his partner has blue eyes?

P(A U B) = P(A) + P(B) - P(A n B) = 108/204 + 114/204 - 78/204 = 70.59%

  1. What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes?

P(A | B) = P(A n B) / P(B) = (78/204) / (114/204) = 0.6842 = 68.42%

  1. What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes? What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?

Probability that a male with brown eyes has a partner with blue eyes = 19/204. Probability that a male has brown eyes = 54/204. (19/204)/(54/204) = 35.19%

Probability that a male with green eyes has a partner with blue eyes = 11/204. Probability that a male has green eyes = 36/204. (11/204)/(36/204) = 30.56%

  1. Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.

The conditional probability of a female with blue eyes provided that the male partner has blue eyes is 78/114.

The probability that a female has blue eyes is 108/204.

Since these probabilities are not equal, it appears that the eye colors of blue eyed respondents are not independent.

It can be generalized that the eye colors of male respondents and their partners are not independent.

Books on a bookshelf. (3.26, p. 114) The table below shows the distribution of books on a bookcase based on whether they are nonfiction or fiction and hardcover or paperback.

  1. Find the probability of drawing a hardcover book first then a paperback fiction book second when drawing without replacement.

(28/95) * (59/94) = 18.5%

  1. Determine the probability of drawing a fiction book first and then a hardcover book second, when drawing without replacement.

(72/95) * (28/94) = 22.58%

  1. Calculate the probability of the scenario in part (b), except this time complete the calculations under the scenario where the first book is placed back on the bookcase before randomly drawing the second book.

(72/95) * (28/95) = 22.34%

  1. The final answers to parts (b) and (c) are very similar. Explain why this is the case.

The only difference in the two calculations is that the second draw in part b has one less total book. This makes for only a minor change in the overall calculations, so the result (final probability) is similar.

Baggage fees. (3.34, p. 124) An airline charges the following baggage fees: $25 for the first bag and $35 for the second. Suppose 54% of passengers have no checked luggage, 34% have one piece of checked luggage and 12% have two pieces. We suppose a negligible portion of people check more than two bags.

  1. Build a probability model, compute the average revenue per passenger, and compute the corresponding standard deviation.

Bags | $ | Pax % 0 | 0 | 54% 1 | 25 | 34% 2 | 35 | 12%

(0 * 0.54) + (25 * 0.34) + (35 * 0.12) = $12.70 per passenger

(0^2 * 0.54) + (25^2 * 0.34) + (35^2 * 0.12) = 359.5

sqrt(359.5 - 12/7^2) = 14.079 stdev

  1. About how much revenue should the airline expect for a flight of 120 passengers? With what standard deviation? Note any assumptions you make and if you think they are justified.

120 * 12.70 = $1524

Income and gender. (3.38, p. 128) The relative frequency table below displays the distribution of annual total personal income (in 2009 inflation-adjusted dollars) for a representative sample of 96,420,486 Americans. These data come from the American Community Survey for 2005-2009. This sample is comprised of 59% males and 41% females.

  1. Describe the distribution of total personal income.

This fata follows the normal distribution.

  1. What is the probability that a randomly chosen US resident makes less than $50,000 per year?

Sum of probabilities from $1 - $49,999 in income = 0.622

  1. What is the probability that a randomly chosen US resident makes less than $50,000 per year and is female? Note any assumptions you make.

0.622 * 0.41 = 0.26 if we assume that sex and income are independent.

  1. The same data source indicates that 71.8% of females make less than $50,000 per year. Use this value to determine whether or not the assumption you made in part (c) is valid.

The assumption is not valid as the same proportion of females should make less than $50,000 per year all respondents (62/2%). Therefore sex and income are dependent.