Suppose you play a game in which two fair standard dice are rolled. If the numbers showing on the dice are different, you lose $2. If the numbers showing are the same, you win $2 plus the dollar value of the sum of dice. Complete the next table, in which the values in the first column are the outcomes for the rolls of the two dice. The second column has the probabilities for those outcomes, and the third column has the payoff values for each outcome. What is the expected value of the game?

Suppose you play the game 36 times. (all possible outcomes)

Your critical outcomes are
1,1 ---expected return = (1/36)(2) + 2 = 2.0555
2,2 ---expected ret;urn = (1/36)(2) + 4 = 4.0555
.
.
6,6 ---expected return = (1/36)(2) + 12 = 12.0555

the expected winnings are 2.0555+4.0555+6.0555+8.0555+10.0555+12.0555
= 42.3333

the other 30 outcomes would all result in a loss of 2
expected losses = (30/36)(2) = 1.6666666

The expected value of the game is $40.67 for 36 game.
So the expected value of one game is 40.67/36 or $1.13

To complete the table, we need to calculate the probabilities and payoff values for each outcome. Let's start by listing all possible outcomes when rolling two fair standard dice:

1. (1, 1)
2. (1, 2)
3. (1, 3)
4. (1, 4)
5. (1, 5)
6. (1, 6)
7. (2, 1)
8. (2, 2)
9. (2, 3)
10. (2, 4)
11. (2, 5)
12. (2, 6)
13. (3, 1)
14. (3, 2)
15. (3, 3)
16. (3, 4)
17. (3, 5)
18. (3, 6)
19. (4, 1)
20. (4, 2)
21. (4, 3)
22. (4, 4)
23. (4, 5)
24. (4, 6)
25. (5, 1)
26. (5, 2)
27. (5, 3)
28. (5, 4)
29. (5, 5)
30. (5, 6)
31. (6, 1)
32. (6, 2)
33. (6, 3)
34. (6, 4)
35. (6, 5)
36. (6, 6)

Now, let's calculate the probabilities for each outcome. Since we have two fair standard dice, the probability of rolling any specific outcome is 1/36 since each die has 6 sides, and there are 6 x 6 = 36 possible outcomes.

Now, let's calculate the payoff values for each outcome based on the rules of the game:

- If the numbers showing on the dice are different, you lose $2.
- If the numbers showing are the same, you win $2 plus the dollar value of the sum of dice.

Using this information, we can complete the table:

Outcome | Probability | Payoff
--------------------------------------
(1, 1) | 1/36 | -2
(1, 2) | 1/36 | -2
(1, 3) | 1/36 | -2
...
(6, 4) | 1/36 | -2
(6, 5) | 1/36 | -2
(6, 6) | 1/36 | 8

To calculate the expected value of the game, we multiply the probability of each outcome by its respective payoff value and sum them all up:

Expected Value = [(1/36)(-2) + (1/36)(-2) + ... + (1/36)(8)].

Let's calculate it:

Expected Value = (-2/36) + (-2/36) + ... + (8/36).

Simplifying,

Expected Value = (-2 - 2 - 2 - 2 - 2 - 2 + 8)/36

Expected Value = (-4 + 8)/36

Expected Value = 4/36

Expected Value = 1/9

Therefore, the expected value of the game is 1/9, which means on average, you'd win $1/9 per game.