You are playing a board game in which the severity of a penalty is determined by rolling three dice and adding the spots on the up-faces. The dice are all balanced so that each face is equally likely, and the three dice fall independently.

(a) Give a sample space for the sum X of the spots.
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20

(b) Find P(X = 6).

(c) If X1, X2, and X3 are the number of spots on the up-faces of the three dice, then X = X1 + X2 + X3. Use this fact to find the mean μX and the standard deviation σX without finding the distribution of X. (Start with the distribution of each of the Xi.)