To find the expected value of the game, we can break it down into the following steps:

1. Identify the possible outcomes of rolling two dice: The sum of the numbers on two dice can range from 2 to 12. Each of these outcomes has a specific probability, which depends on how many combinations of the dice can result in that sum.

2. List the sums and their probabilities:

   • The possible sums (2 through 12) occur with the following probabilities:

   $\text{Sum} \quad \text{Number of Ways} \quad \text{Probability}$ $2 \quad \quad 1 \quad \quad \frac{1}{36}$ $3 \quad \quad 2 \quad \quad \frac{2}{36} = \frac{1}{18}$ $4 \quad \quad 3 \quad \quad \frac{3}{36} = \frac{1}{12}$ $5 \quad \quad 4 \quad \quad \frac{4}{36} = \frac{1}{9}$ $6 \quad \quad 5 \quad \quad \frac{5}{36}$ $7 \quad \quad 6 \quad \quad \frac{6}{36} = \frac{1}{6}$ $8 \quad \quad 5 \quad \quad \frac{5}{36}$ $9 \quad \quad 4 \quad \quad \frac{4}{36} = \frac{1}{9}$ $10 \quad \quad 3 \quad \quad \frac{3}{36} = \frac{1}{12}$ $11 \quad \quad 2 \quad \quad \frac{2}{36} = \frac{1}{18}$ $12 \quad \quad 1 \quad \quad \frac{1}{36}$

3. Calculate the expected value: The expected value is the sum of each possible outcome multiplied by its probability. Let’s calculate it:

$E(\text{Sum}) = 2 \times \frac{1}{36} + 3 \times \frac{1}{18} + 4 \times \frac{1}{12} + 5 \times \frac{1}{9} + 6 \times \frac{5}{36} + 7 \times \frac{1}{6} + 8 \times \frac{5}{36} + 9 \times \frac{1}{9} + 10 \times \frac{1}{12} + 11 \times \frac{1}{18} + 12 \times \frac{1}{36}$

Let me compute this sum.The expected value of the sum of the numbers that appear on the dice is approximately $7.00$.

4. Calculate the expected net value:

   • You pay $8 to play the game.
   • Your expected payoff is $7.00 (the expected value of the dice roll).

   Therefore, the expected net value is: $\text{Expected net value} = 7.00 - 8 = -1.00$

The expected value of the game is $-1.00$ dollars, meaning you are expected to lose $1.00 on average per game.

Would you like more details or have any questions?

Here are five related questions you might find useful:

1. What is the expected value of a game where the payoff is doubled?
2. How does the probability distribution of sums change if one die is biased?
3. What is the variance of the sum of two dice rolls?
4. Can you find the expected value of rolling three dice instead of two?
5. How would the game change if the entry cost was $5 instead of $8?

Tip: Expected value gives you an idea of long-term outcomes but doesn't guarantee the result of any single trial.