(A) Suppose that a fair die is rolled 100 times. What is the probability of throwing exactly twenty fives?

> dbinom(20, 100, 1/6)
[1] 0.067862
> 

X counts number of fives thrown
X follows binomial distribution
20 successes out of 100 trials, each with success probability 1/6


(B) Suppose the height of men in Wisconsin is known to be normally distributed with a mean of
175 centimeters and a standard deviation of 12 centimeters. If you were to select a man from the
Wisconsin population at random, what is the probability that he would be more than 180 cm? Less
than 160 cm?

mean = 175, sd = 12 normal distribution (not standard normal)

P(>180cm):
> pnorm(180, mean=175, sd=12, lower.tail=F)
[1] 0.3384611
> 

P(<160cm):
> pnorm(160, mean=175, sd=12)
[1] 0.1056498
> 


(C) If you were to roll a fair die 200 times, what is the probability of rolling a six no more than 10
times? More than 45 times?

no more than 10 times, is the cumulative probability of up to 10 or the sum of rolling zero, one, two, .., up to 10 sixes

P(six no more than 10 times)
> pbinom(10, 200, 1/6)
[1] 4.486632e-07
> 

P(six more than 45 times)
> pbinom(45, 200, 1/6, lower.tail=F)
[1] 0.01283258
> 
can also evaluate via complement event.


(D) Malfunctions in a laptop computer brand are known to follow an exponential distribution with
a mean time of 36 months until the device malfunctions. What is the probability that a randomly
selected device will malfunction within the first 12 months?

Here X is exponential r.v. counts number of months till the computer malfunctions

> pexp(12, rate=1/36)
[1] 0.2834687
>


(E) Suppose there are ten multiple choice questions on a Stats exam. Each question has four
possible answers, and only one of them is correct. Find the probability of having six or less correct
answers if a student attempts to answer every question at random.

X counts # of correct answers
X follows a binomial distribution. sucess probability = 1/4
P(less than or equal 6 correct) = sum(P(i correct)) for i from 0 to 6
which can be evaluated as follows:

> pbinom(6, 10, 1/4)
[1] 0.9964943
>

Can also add dbinom(0, size=10, prob=1/4) + ... + dbinom(6, size=10, prob=1/4) to get the same answer.