1.

(3.41) or Property F

2.

P(W = 6) = P(X=4,Y=1 or X=2,Y=2) = 1/36 + 1/36

3.  

p(1-p) where p = 2/36 from Problem 2

4.

libsim <- function(nreps) {
   prt <- sample(c(4,5,6,7),nreps,replace=T,prob=c(0.1,0.2,0.3,0.4))
   i <- sample(c(0,1),nreps,replace=T)
   b <- prt + 2*i
   mean(b)
}

5.

Write X as X1+X2+...+X10, where Xi is the indicator variable of the
event that an HTH pattern starts with the i-th toss.  Then

EX = EX1 + ... + EX10 = 10 P(Xi = 1) = 10 (1/2)^3 = 1.25