2.6 Dice rolls

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

a) getting a sum of 1?

The probability is 0, because the minimum sum is 2.

b) getting a sum of 5?

The set is = {(1,4), (4,1), (2,3), (3,2)} out of 36 possible outcomes. Therefore the probabiity is 4/36 = 1/9.

c) getting a sum of 12?

There is one set satisfy this {(6,6)}. Therefore the probability is 1/36.

2.8 Poverty and language

(a)

No, they are not disjoint. 4.2% of the sample in the survey fall into both categories.

(b)

https://imgur.com/PsztCag

(c)

below_pov <- 14.6/100
speak_foreign <- 20.7/100
both <- 4.2/100

0.104 of Americans live below poverty and only speak English at home.

(d)

0.311 of Americans live below poverty or speak foreign language at home.

(e)

0.896 of Americans live above poverty and only speak English at home.

(f)

No. They are dependent because probability of below poverty given speaking foreign language is 4.2%/20.7% = 0.2028986, which is not the same as probabiility of below proverty line. By Baye’s theorem, they must be depenent.

2.20 Assortative mating

(a)

(114+108-78)/204
## [1] 0.7058824

(b)

78/114
## [1] 0.6842105

(c)

19/54
## [1] 0.3518519
11/36
## [1] 0.3055556

(d)

No, the eye colors are dependent. It is apparent that the male repsondents and their partners tend to choose the person with same eye color as partner.

2.30 Books on a bookshelf

(a)

13/95*(59/94) + 15/95*(59/94)
## [1] 0.1849944

(b)

59/95*(28/94) + 13/95*(27/94)
## [1] 0.2243001

(C)

59/95*(28/95) + 13/95*(28/95) 
## [1] 0.2233795

(d)

They are the similar because there are 95 books. The replacement of one book does not matter that much, comparing 1/95 = 0.0105263 and 1/94 = 0.0106383.

2.38 Baggage fees

(a)

The average revenue per passenger is 25x34%+(35+25)x12% = 15.7. The variance is (25^2) x 35% + (60^2) x 12% - (25x34%+60x12%)^2 = 398.01 The standard deviation is square root of variance = 19.950188

(b)

The expected revenue and standard deviation of that revenue are:

120 * (25*0.34 + (35+25)*0.12)
## [1] 1884
120^0.5 * (((25^2)*0.34+(60^2)*0.12 - (25*0.34+60*0.12)^2)^0.5)
## [1] 218.5434

2.44 Income and gender

(a)

The income is skewed to the right. About 62% of the sample make below $49,999 a year. So the median income is between $34,999 and $49,999. Very few make less than $9,999 or higher than $100,000

(b)

0.622

(c)

This is 41% x 62.2% = 0.25502

(d)

In (c) I assume that income and gender are independent. This is now invalid, knowing that 71.8% of females make less than 50k a year.