W2_606

Chapter 2 - Probability Graded: 2.6, 2.8, 2.20, 2.30, 2.38, 2.44

library('VennDiagram')

2.6 If you roll a pair of fair dice, what is the probability of

getting a sum of 1?
```
s={}
P(s)=0
```

getting a sum of 5?

s={(1,4),(4,1),(2,3),(3,2)}
p(s)=4/36 = 1/8

getting a sum of 12?
```
s={(6,6)}
p(s)=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

pr_poor<-14.6/100 # %Americans live below the poverty line
pr_poor

[1] 0.146

pr_ESL<-20.7/100 # %speak a language other than English
pr_ESL

[1] 0.207

pr_poorESL<-4.2/100 # %fall into both categories
pr_poorESL

[1] 0.042

Are living below the poverty line and speaking a foreign language at home disjoint?
```
No.
```
Draw a Venn diagram summarizing the variables and their associated probabilities.

venn.plot <- draw.pairwise.venn(area1 = pr_poor,area2 = pr_ESL,cross.area = pr_poorESL,category = c("% Poor", "% ESL"),fill = c("light blue", "pink"));
grid.draw(venn.plot);

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

Ans<-pr_poor-pr_poorESL
Ans

[1] 0.104

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

Ans<-pr_poor+pr_ESL-pr_poorESL
Ans

## [1] 0.311

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

Ans<-1-(pr_poor+pr_ESL-pr_poorESL)
Ans

[1] 0.689

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

Ans<-pr_poor-pr_poorESL
Ans

[1] 0.104

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.

Partner (female)

##	Blue	Brown	Green	Total
Blue	78	23	13	114
Brown	19	23	12	54
Green	11	9	16	36
Total	108	55	41	204

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

ans<-108/204
ans

[1] 0.5294118

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

ans<-78/204
ans

[1] 0.3823529

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

ans<-19/204
ans

[1] 0.09313725

What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?

ans<-11/204
ans

[1] 0.05392157

Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.
```
No. Eye color of two male and female variable are joined together, female's eye color is depent on male's.
```

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.

Format

##	Hardcover	Paperback	Total
Fiction	13	59	72
Nonfiction	15	8	23
Total	28	67	95

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

ans<-(28/95)*(59/94)
ans

[1] 0.1849944

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

ans<-(72/95)*(28/94)+(72/95)*(27/94)
ans

[1] 0.443449

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.

ans<-(72/95)*(28/95)+(72/95)*(27/95)
ans

[1] 0.4387812

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

The difference is part b withou replacement and part c with replacement.With replacement, the probability of second object is picked also base on the complete data set.

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

avg<-25*0.34+(35+25)*0.12
avg

[1] 15.7

var_x<-(0-avg)^2*0.54+0.34*(25-avg)^2+0.12*(35+25-avg)^2
standard_dev<-sqrt(var_x)
standard_dev

[1] 19.95019

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.

#E(120x)=120*E(x)
E<-120*avg
E

[1] 1884

standard_dev<-sqrt(0.54*(0-E)^2+0.34*(25*120-E)^2+0.12*(60*120-E)^2) #35+25=60 for people who has tow pieces.
standard_dev

[1] 2394.023

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.

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%

Describe the distribution of total personal income.

The distribution is nearly normal, and mean is very closed to median.

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

ans<-0.022+0.047+0.158+0.183+0.212
ans

[1] 0.622

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.

undefine
We don't know the female or male 's saraly distribution. If data distribution is skew, the general probability can not represent one of female/male distribution.

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.

#My assumption in c is correct. The femail's saraly is left skew. 
ans<-0.718*0.41
ans

## [1] 0.29438

W2_606

Chunhui Zhu

September 10, 2017

Partner (female)

Format