If you roll a pair of fair dice, what is the probability of
The probability of getting a sum of one when rolling a pair of fair dice is 0 since the minimum sum would be 2 if you rolled two 1’s.
The probability of getting a sum of 5 when rolling a pair of fair dice is \(\frac{4}{36}\) since there are 4 ways you can get a sum of 5 (1, 4), (2, 3), (3, 2), and (4, 1) out of 36 possible rolls.
# There are 6 times 6 combinations possible when rolling two 6-sided dice
6*6## [1] 36The probability of getting a sum of 12 when rolling a pair of fair dice is \(\frac{1}{36}\) since there is only one way to get a sum of 12 if you rolled two 6’s.
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.
No, since there are 4.2% of the population that fall into both categories, living below the poverty line, and speaking a language other than English at home are not disjoint.
10.4% of Americans (14.6% minus 4.2%) live below the poverty line and only speak English at home.
14.6 - 4.2## [1] 10.431.1% of Americans live below the poverty line or speak a foreign language at home.
14.6 + 20.7 - 4.2## [1] 31.168.9% of Americans live above the poverty line and only speak English at home.
100 - 14.6 - (20.7 - 4.2)## [1] 68.9If the two events were independent then knowing the outcome of one should not affect the probability of the second, but the probability of living below the poverty line is significantly higher when you know that they speak a language other than English at home. As we saw above 10.4 percent of Americans live below the poverty line and only speak English at home and we know that 4.2 percent live below the poverty line and speak a foreign language at home. Knowing that you speak a foreign language at home means there is a 20.2% probability that live below the poverty line wherease if you do not speak a foreign language at home, your probability of living below the poverty line is only 13.1%.
4.2/20.7## [1] 0.202898610.4/(100-20.7)## [1] 0.1311475Assortative 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.
Partner (female) runs across horizontally and Self (male) runs vertically
| — | Blue | Brown | Green | Total |
|---|---|---|---|---|
| Blue | 78 | 23 | 13 | 114 |
| Brown | 19 | 23 | 12 | 54 |
| Green | 11 | 9 | 16 | 36 |
| Total | 108 | 55 | 41 | 204 |
# probability that a randomly chosen male respondent or his partner has blue eyes
(108 + 114 - 78)/204## [1] 0.7058824# probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes
78/114## [1] 0.6842105# probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes
19/54## [1] 0.3518519# probability of a randomly chosen male respondent with green eyes has a partner with blue eyes
11/36## [1] 0.3055556No, it appears that male respondents eye color and their partner’s eye color are not independent. The probability of a partner’s eye color being the same as their own is higher than the probability of any other color in all cases.
# probability that a randomly chosen male respondent with brown eyes has a partner with brown eyes
23/54## [1] 0.4259259# probability of a randomly chosen male respondent with green eyes has a partner with green eyes
16/36## [1] 0.4444444The table below shows the distribution of books on a bookcase based on whether they are nonfiction or fiction and hardcover or paperback.
| — | Hardcover | Paperback | Total |
|---|---|---|---|
| Fiction | 13 | 59 | 72 |
| Nonfiction | 15 | 8 | 23 |
| Total | 28 | 67 | 95 |
28/95 * 59/94## [1] 0.1849944# if the first book was a hardcover fiction book
72/95 * 27/94## [1] 0.2176932# if the first book was a paperback fiction book
72/95 * 28/94## [1] 0.2257559# if the first book was a hardcover fiction book
72/95 * 28/95## [1] 0.2233795From the textbook page 103: “When the sample size is only a small fraction of the population (under 10%), observations are nearly independent even when sampling without replacement.” In this case our sample is only 2/95 or about 2% of our population.
2/95## [1] 0.02105263An 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.
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.
| Income | Total |
|---|---|
| $1 to $9,999 or loss | 2.2% |
| $10,000 to $14,999 | 4.7% |
| $15,000 to $24,999 | 15.8% |
| $25,000 to $34,999 | 18.3% |
| $35,000 to $49,999 | 21.2% |
| $50,000 to $64,999 | 13.9% |
| $65,000 to $74,999 | 5.8% |
| $75,000 to $99,999 | 8.4% |
| $100,000 or more | 9.7% |
The distribution seems to be slightly right skewed with the mode below the halfway point.
lessthan50k <- (21.2 + 18.3 + 15.8 + 4.7 + 2.2)
lessthan50k## [1] 62.2Assuming that the proportion of males and females in each income bracket are the same as the proportion of males to females in the overall population the probability that a randomly chosen US resident makes less than $50,000 per year and is female is 25.5%.
lessthan50k * .41## [1] 25.502Since the overall proportion of the population who make less than $50,000 per year is 62.2% we would expect that if the proprtions of males and females at each income bracket were the same as the population as a whole then 62.2 percent of women should make below $50,000 per year. So my assumption above is not valid.