Question

We roll a pair of dice. If the sum of the dice is 7, you pay me $26. If the sum is not 7, I pay you the number of dollars indicated by the sum of the dice. What is your expected value for the game?

Answer 1

Let's make a table of all of the ways you can roll the various numbers:

2: 11
3: 12 21
4: 13 22 31
5: 14 23 32 41
6: 15 24 33 42 51
7: 16 25 34 43 52 61
8: 26 35 44 53 62
9: 36 45 54 63
10: 46 55 64
11: 56 65
12: 66

There are 6 ways out of 36 different rolls to roll a 7. 6/36 * ($26) = ($4.33)
There is 1 way out of 36 different rolls to roll a 2, which pays $2: 1/36*$2 = $0.06
There are 2 ways out of 36 rolls to roll a 3, paying $3: 2/36*$3 = $0.17
etc.
Add them all up (don't forget the $4.33 in parentheses is a negative amount) and that should be your EV.