1. Pr[X=11] = 2/36. The sample space is all 36 pairs (i,j) where i
represents the number that came up on the first die and j the number
on the second die. The only outcomes where X=11 is true are these
two: (5,6) and (6,5).
2. Pr[W=1] = Pr[X=7 or X=11] = Pr[X=7] + Pr[X=11] = 6/36 + 2/36 = 8/36.
Here we have used the fact that the events X=7 and X=11 are disjoint
(a random variable can't both be equal to 7 and equal to 11 at the same
time!), and that there are 6 outcomes where X=7 is true, namely,
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
3. We saw above that Pr[W=1] = 8/36. Also Pr[W=-1] = Pr[X=2] + Pr[X=3]
+ Pr[X=12] = 1/36 + 2/36 + 1/36 = 4/36. Now calculate:
E[W] = 1*Pr[W=1] + 0*Pr[W=0] + (-1)*Pr[W=-1]
= 8/36 - 4/36 = 4/36.
So the final answer is 4/36. This is a good game for you, and a very
poor game for the casino!