Suppose that you toss a fair six-sided {1,2,3,4,5,6} die and let X represent the outcome obtained. Find the expected volume of a box that has a height of 24 inches and a square base with side length X inches.

The expected volume of the box is 364 cubic inches.

Step-by-step explanation:

Since the die is fair, then P (X=k) = 1/6 for any k in {1,2,3,4,5,6}. Let Y represent the volume of the box in cubic inches. For how the box is formed, Y = X²*24. Thus, the value of Y depends directly on the value of X, and we have

(When X = 1) Y = 1²*24 = 24, with probability 1/6 (the same than P (X=1) (When X = 2) Y = 2²*24 = 96, with probability 1/6 (the same than P (X=2) (When X = 3) Y = 3²*24 = 216, with probability 1/6 (the same than P (X=3) (When X = 4) Y = 4²*24 = 384, with probability 1/6 (the same than P (X=4) (When X = 5) Y = 5²*24 = 600, with probability 1/6 (the same than P (X=5) (When X = 6) Y = 6²*24 = 864, with probability 1/6 (the same than P (X=6)

As a consequence, the expected volume of the box in cubic inches is

E (Y) = 1/6 * 24 + 1/6*96 + 1/6*216 + 1/6*384 + 1/6*600+1/6*864 = 364