Two dice are rolled and the product of the two numbers is calculated. What is the expected value of a roll?

Answer 1

The expected roll is just the weighted average of all the outcomes.

1,1 --> 1
1,2 --> 2
1,3 --> 3
1,4 --> 4
1,5 --> 5
1,6 --> 6
2,1 --> 2
2,2 --> 4
...
5,6 --> 30
6,1 --> 6
6,2 --> 12
6,3 --> 18
6,4 --> 24
6,5 --> 30
6,6 --> 36

Add these all up and divide by 36. That's your expected value of a roll.

1 + 2 + 3 + 4 + 5 + 6 = 21
2 + 4 + 6 + 8 + 10 + 12 = 42
3 + 6 + ... + 18 = 63
4 + 8 + ... + 24 = 84
5 + 10 + ... + 30 = 105
6 + 12 + ... + 36 = 126

21 + 42 + 63 + 84 + 105 + 126 = 441

441 / 36 = 12.25

So the expected value is 12.25

Answer 2

Just take all the possibilities and add them up:

1,1 = 1
1,2 = 2
1,3 = 3
...
6,4=24
6,5=30
6,6=36

The formula then look like this:
1(1+2+3+4+5+6) +
2(1+2+3+4+5+6) +
...
6(1+2+3+4+5+6)

Factor the big sum into:
(1+2+3+4+5+6)*(1+2+3+4+5+6)

And you get
21*21 = 441

441/36 different combinations =
12.25

expected value = 12.25

Answer 3

6 or 12

there is a 1/9 chance of rolling either