2.6

a)Probability would be zero as there is no possible outcome for two fair dice to get a sum of 1 b)There are four possible combinations that could give a sum of 5: (2,3) , (3.2), (4,1) , (1,4). The maximum amount of outcomes is 36, as per 6 possible outcomes per dice.

4/36
## [1] 0.1111111

c)To get a sum of two you must throw two sixes which means there are only two possible combinations.

2/36
## [1] 0.05555556

2.12

# Let A = event that kid misses one day
# Let B = event that kid misses two days
# Let C =  event that kid misses three or more days
pa <- .25
pna <- 1-pa
pb <- .15
pnb <- 1-pb
pc <- .28
pnc <- 1-pc
  1. What is the probability that a student chosen at random doesn’t miss any days of school due to sickness this year?
1- (pa+pb+pc)
## [1] 0.32
  1. What is the probability that a student chosen at random misses no more than one day?
1- (pb+pc)
## [1] 0.57
  1. What is the probability that a student chosen at random misses at least one day?
pa+pb+pc
## [1] 0.68
  1. If a parent has two kids at a DeKalb County elementary school, what is the probability that neither kid will miss any school? Note any assumption you must make to answer this question. Assuming that they are independent events we will use the same number as per question a
{1- (pa+pb+pc)}*{1- (pa+pb+pc)}
## [1] 0.1024
  1. If a parent has two kids at a DeKalb County elementary school, what is the probability that both kids will miss some school, i.e. at least one day? Note any assumption you make. Assuming that they are independent events we will use the same number as per question c
(pa+pb+pc)*(pa+pb+pc)
## [1] 0.4624
  1. If you made an assumption in part (d) or (e), do you think it was reasonable? If you didn’t make any assumptions, double check your earlier answers.

I believe it was reasonable because it is safe to assume that when one child gets sick in the house he/she is kept separate from the healthy child as to avoid spreading the sickness. We also did not have numerical data as to make an assumption including the probability of the contagion of disease.

2.18

  1. Are being in excellent health and having health coverage mutually exclusive? No they are not because the intersection between both events does not equal to zero

  2. What is the probability that a randomly chosen individual has excellent health?

# Let E = excellent health

pe <- 0.2327
pne <- 1-pe

The probability is 0.2327

  1. What is the probability that a randomly chosen individual has excellent health given that he has health coverage?
# Let H = health insurance
ph <- 0.8738
pnh <- 1-ph
pewh <- 0.2099
pegh <- pewh/ph
pewh/ph
## [1] 0.2402152
  1. What is the probability that a randomly chosen individual has excellent health given that he doesn’t have health coverage?
pewnh <- 0.0230
pegnh <- pewnh/pnh
pewnh/pnh
## [1] 0.1822504
  1. Do having excellent health and having health coverage appear to be independent? Because a person’s coverage information helps predict whether a person has excellent health or not (since the data shows those with health coverage have a drastically higher amount of people in execellent health), one can say the variables are dependant.

2. 24

## Let S = votes in favor of Scott
## Let G = voters for Scott graduated from college
## Let F = voters against Scott that graduated from college
ps <- 0.53
pns <- 1-ps   #voters against Scott
pg <- 0.37    #voters in favor of Scott with college education
png <- 1-pg
pf <- 0.44  #voters against Scott with college education
pnf <- 1-pf 
(ps*pg)+(pns*pf) #Voters with a college degree with or against Scott
## [1] 0.4029
pce <- 0.4029 
psgg <-(ps*pg) #Given that voted for Scott voter is college educated
psgce <- {(ps*pg)/pce}
{(ps*pg)/pce}
## [1] 0.4867213

The probability that any college educated voter voted for Scott was 0.4867

2.30

  1. Find the probability of drawing a hardcover book first then a paperback fiction book second when drawing without replacement.
# Let HB = Hardcover book
# Let PF = Paperback Fiction No Replacement
phb <- 28/95
ppf <- 59/94
phbapf <- phb*ppf 
phb*ppf
## [1] 0.1849944
  1. Determine the probability of drawing a fiction book first and then a hardcover book second, when drawing without replacement.
# Let FB = Fiction Book
# Let HBNR = Hardcover Book No Replacement
pfb <- 72/95 
phbnr <- 28/94
pfbahbnr <- pfb*phbnr
pfb*phbnr
## [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.
pfb*phb
## [1] 0.2233795
  1. The final answers to parts (b) and (c) are very similar. Explain why this is the case. Because the difference between the total number in the denominator only changes by one, thus making the difference small specially as numbers become bigger.

2.36

  1. Create a probability model and find Andy’s expected profit per game.
pnc <- (52-12-4)/52 #Number card
pfc <- (12/52) #Face card
pac <- (4/52) #Ace card
pacc <- (1/52) #Ace of Clubs card
-2+(pnc*0)+(pfc*3)+(pac*5)+(pacc*20)
## [1] -0.5384615
  1. Would you recommend this game to Andy as a good way to make money? Explain. No because the he has a negative probability of slightly over 50% meaning that not only would he not win but he would lose more money than he earns.

2.42 Ice cream usually comes in 1.5 quart boxes (48 fluid ounces), and ice cream scoops hold about 2 ounces. However, there is some variability in the amount of ice cream in a box as well as the amount of ice cream scooped out. We represent the amount of ice cream in the box as X and the amount scooped out as Y . Suppose these random variables have the following means, standard deviations, and variances:

Completely lost on this one I tried to look online for help but answers and methods were very very different.

  1. An entire box of ice cream, plus 3 scoops from a second box is served at a party. How much ice cream do you expect to have been served at this party? What is the standard deviation of the amount of ice cream served?

  2. How much ice cream would you expect to be left in the box after scooping out one scoop of ice cream? That is, find the expected value of X Y. What is the standard deviation of the amount left in the box?

  1. Using the context of this exercise, explain why we add variances when we subtract one random variable from another.