Bill and Fred are betting on the outcome when two fair dice are rolled. If the dice sum is any even number, other than 8, Bill pays Fred $8. If the dice sum is 8, Bill pays Fred $1. If the dice sum is odd, Fred pays Bill $2. What is Fred's expected payoff for this game?

P(even,~8) = P(2,4,6,10,12)

= 1/36 + 3/36 + 5/36 + 3/36 + 1/36 = 13/36
P(8) = 5/36
P(odd) = 1-(13/36 + 5/36) = 1/2

So, Fred gets 8*13/36 + 1*5/36 - 2*1/2 = 73/36, or about $2

To find Fred's expected payoff for this game, we need to calculate the probability of each outcome and multiply it by the payoff associated with that outcome. We can then sum up all the expected payoffs to get the overall expected payoff for Fred.

Let's start by calculating the probabilities of each outcome:

1. When the dice sum is any even number, other than 8:
- There are 18 different combinations of outcomes that result in an even sum on two dice (2, 4, 6, 8, 10, 12).
- Out of these 18 combinations, 3 of them result in a sum of 8 (3, 5, 7) which would be dealt with separately.
- So, there are a total of 18 - 3 = 15 combinations that result in an even sum, other than 8.
- The probability of rolling one of these combinations is 15/36 = 5/12, since there are 36 possible outcomes from rolling two dice.

2. When the dice sum is 8:
- There are 5 different combinations of outcomes that result in a sum of 8 (2+6, 3+5, 4+4, 5+3, 6+2).
- The probability of rolling one of these combinations is 5/36.

3. When the dice sum is odd:
- Similar to the even number case, there are 18 different combinations of outcomes that result in an odd sum on two dice.
- The probability of rolling one of these combinations is 18/36 = 1/2.

Now let's calculate the expected payoff for each outcome:

1. For even sums, other than 8:
- Bill pays Fred $8, so the payoff is -$8 for Fred since it's a loss.
- The probability of this outcome is 5/12.
- The expected payoff for this outcome is (-$8) * (5/12) = -$40/12.

2. For a sum of 8:
- Bill pays Fred $1, so the payoff is -$1 for Fred since it's a loss.
- The probability of this outcome is 5/36.
- The expected payoff for this outcome is (-$1) * (5/36) = -$5/36.

3. For odd sums:
- Fred pays Bill $2, so the payoff is +$2 for Fred since it's a gain.
- The probability of this outcome is 1/2.
- The expected payoff for this outcome is (+$2) * (1/2) = +$2/2 = +$1.

Now let's sum up the expected payoffs for all outcomes:

(-$40/12) + (-$5/36) + (+$1) = (-120/36) + (-5/36) + (+1) = -125/36 + 36/36 + 36/36 = -125/36 + 72/36 + 36/36 = -17/36 + 1 + 1 = -17/36 + 36/36 = 19/36.

Therefore, Fred's expected payoff for this game is +19/36 (approximately +$0.528).