Question

Consider a game where you roll a six-sided die and a four-sided die, then you subtract the number on the four-sided die from the number on the six-sided die. If the number is positive, you receive that much money (in dollars). If the number is negative, you pay that much money (in dollars).

For example, you might roll a 5 on the six-sided die and a 2 on the four-sided die, in which case you would win $3. You might instead roll a 1 on the six-sided die and a 4 on the four-sided die, in which case you would lose $3.

What is the expected value of this game, in dollars?

Do not put a dollar sign ($). Give your answer to the nearest penny.

Answer 1

Sample space with payoff (in red) is shown below:

Die 2 1 2 3 4 1 2 3 Die 1 (1,1) 0 (2,1) 3 (3,1) 3 (4,1) 3 (5,1) 3 (6,1) 3 (1,2) (1,3) (1,4) -3 -3 -3 (2,2) (2,3) (2,4) 0 -3 -3 (3,2) (3,3) (3,4) 3 0 -3 (4,2) (4,3) (4,4) 3 3 0 (5,2) (5,3) (5,4) 3 3 3 (6,2) (6,3) (6,4) 3 3 3 4 5 6

Out of 24 outcomes, you win 14 times, you lose 6 times and tie 4 times.

The payoff table is shown below:

x	P(x)
3	14/24
-3	6/24
0	4/24

The expected value is:

$E(X)=\sum xP(x)=3(14/24)-3(6/24)+0(4/24)={\color{Red} \$1.75}$