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 - two dice will add up to a minimum of 2

  1. getting a sum of 5?

4 cases that can net this result hence probability = 4/36 or .1111

  1. getting a sum of 12?

only a single case can net 12 hence probabilty= 1/36

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?

4.2 % of the people fall into this category so no.

  1. Draw a Venn diagram summarizing the variables and their associated probabilities.

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

14.6 - 4.2 
## [1] 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 
## [1] 31.1
  1. What percent of Americans live above the poverty line and only speak English at home?
100 - 31.1
## [1] 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 one speaks a foreign language there is a 20.2% probability that they will live below the poverty line as opposed to 13.1% when one speaks english

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?
(108 + 114 - 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. Whatistheprobabilitythatarandomlychosenmalerespondentwithbrowneyeshasapartner with blue eyes? What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?
# brown vs blue eyed partner
19/54
## [1] 0.3518519
#green vs blue eyed partner
11/36
## [1] 0.3055556
  1. Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.

Eye colors do not seem independent. probabilities calculated above are different and show that the values 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) 
## [1] 0.1849944
  1. Determine the probability of drawing a fiction book first and then a hardcover book second, when drawing without replacement.
(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 very small (approx 2%) as such even without replacement samples are independent

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.
hist(c(2.2,4.7,15.8,18.3,21.2,13.9,5.8,8.4,9.7))

The distibution seems bi modal with a negative skew

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

If we assume that the ratio of men vs women remains consistent then the probability will be

6220 * .41 
## [1] 2550.2

Note any assumptions you make. (d) 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 probability of picking a woman making less than 50k is 29.4% if keeping to the assumption. As such, the assumption is incorrect