Q1:

# P(D = 3) = P(N = 8 or N = 14) 
print((1-0.15)^7 * 0.15 + (1-0.15)^13 * 0.15)

Q2:

parksim <- function(nreps) {
   nvals <- rgeom(nreps,0.15) + 1
   dvals <- ifelse(nvals <= 10,11-nvals,nvals-11)
   mean(dvals == 3)
}

print(parksim(10000))

Q3:

# P(G_i = 1) is still 6/9!
print(2*4*(6/9) - 4)

Q4:

varxy <- function(v,w) 
{
   # Cov(X,Y) = E(X*Y) - EX * EY = P(X=Y=1) - v*v = w - v^2
   2*v*(1-v) + 2*(w - v^2)
}

print(varxy(0.5,0.2))

Q5:

calc1short <- function(n,s)  
{
   r <- 3
   n22 <- n^2 / 2
   num <- choose(n-3,r-2) * choose(n22-5*n/2+3,s-(r-2))
   den <- choose(n22-3*n/2,s)
   num/den
}

print(calc1short(80,20))  # about 0.31