2.6 Dice rolls. If you roll a pair of fair dice, what is the probability of:

  1. getting a sum of 1?

Zero. The min sum that can be is 2.

  1. getting a sum of 5?

Possible combinations: 1:4, 4:1, 2:3, 3:2. Hence, 4/36.

  1. getting a sum of 12?

Possible combination: 6:6. Hence, 1/36.

2.8 Poverty and language. 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.59

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

No, as it can happen at the same time. It is said: ‘4.2% fall into both categories’

  1. Draw a Venn diagram summarizing the variables and their associated probabilities.
# install.packages('VennDiagram')
library(VennDiagram)
## Loading required package: grid
## Loading required package: futile.logger
grid.newpage()
draw.pairwise.venn(area1 = 0.1, area2 =0.165, cross.area = 0.042, category = c("Poverty", "Foreign language speakers"), lty = rep("blank",
    2), fill = c("light blue", "pink"),alpha = rep(0.5, 2), cat.pos = c(0,0), cat.dist = rep(0.025, 2))

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

14.6% - 4.2% = 10.4%

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

14.6% + 20.7% - 4.2% = 31.1%

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

100% - (14.6% + 20.7% -4.2%) = 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?

Yes, two events are independent as knowing the outcome of one provides no useful information about the outcome of the other.

2.20 Assortative mating. 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?
114/204 + 108/204 - 78/204 
## [1] 0.7058824
  1. What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes?
78/114 
## [1] 0.6842105
  1. What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes?
19/54 
## [1] 0.3518519
  1. What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?
11/36 
## [1] 0.3055556
  1. Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.

When two events are independent, we can say that P(A and B)=P(A)⋅P(B)

Blue+Blue:

(78/204) = (114/204) * (108/204) This equation is not true, hence these events are not independent.

2.30 Books on a bookshelf. 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)
## [1] 0.1849944
  1. Determine the probability of drawing a fiction book first and then a hardcover book second, when drawing without replacement.

If the first book is a hardcover fiction book:

(72/95) * (27/94)
## [1] 0.2176932

If was a paperback fiction book first:

(72/95) * (28/94)
## [1] 0.2257559
  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)
## [1] 0.2233795
  1. The final answers to parts (b) and (c) are very similar. Explain why this is the case.

The sample size is quite large, if we take one book out of 94 it will not affect the result significantly.

2.38 Baggage fees. 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.
fees<-c(0, 25, 60)
prob<-c(0.54, 0.34, 0.12)
df<-data.frame(fees,prob)
EV<-sum(df$fees*df$prob)
df$var<-((df$fees-EV)**2)*df$prob
std<-sqrt(sum(df$var))
df
##   fees prob      var
## 1    0 0.54 133.1046
## 2   25 0.34  29.4066
## 3   60 0.12 235.4988
EV
## [1] 15.7
std
## [1] 19.95019

Average revenue per passenger is $15.7, standard deviation is $19.95

  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.

Expected Revenue for a flight of 120 passenger:

120*15.7
## [1] 1884

Standard deviation for a flight of 120 passenger:

120*19.95
## [1] 2394

Expected revenue for 120 passangers: $1884 with std $2394 assuming 54% of passengers have no checked luggage, 34% have one piece of checked luggage, 12% have two pieces and negligible portion of people check more than two bags.

2.44 Income and gender. 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.
income <- c(0.022,0.047,0.158,0.183,0.212,0.139,0.058,0.084,0.097)
barplot(income)

The distribution is skewed to the right. Pick is at $35,000 to $49,999 income (21.2%)

  1. What is the probability that a randomly chosen US resident makes less than $50,000 per year?
2.2 + 4.7 + 15.8 + 18.3 + 21.2
## [1] 62.2
  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.
62.2*0.41
## [1] 25.502

Assumption: gender 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.

71.8% is not equal to 25.5%

I can conclude that the assumption that I have made in part (c) is not valid and these events are dependent.