Non-Poker Probability Question (somewhat OT)

[Marcotte]

This post is probably better suited to the Probability forum, but I read/post here much more so here goes.

I want to calculate the EV of the following situation. I will roll X dice, each with Y sides. The result of the roll will be the highest face showing.

Example: I roll five (5) six-sided dice. (5d6). I roll a 1, 2, 2, 3, 4. The result is 4.

I roll 2 six-sided dice. (2d6). I roll a 3 and a 6. Result is six.

I'm looking for a general answer, as I want to compare EVs of many different combinations (1d2, 2d2...5d2; 1d3, 2d3...5d3; 1d4, 2d4...5d4; etc.)

Bonus question.
Instead of taking only the highest number rolled as result, take all max rolls plus the next highest roll and add them together. So for 5d6, with a roll of 1,3,5,6,6, you get 17 (6+6+5). Now what is the EV?

[bigredlemon]

how can you calculate EV without stating the wager? What are you betting for? What is a win, and what is a loss?

[Marcotte]

Sorry, should have said expected outcome. I'm looking for the value of the roll. There is no betting involved.

If I roll one die (keeping it all six-sided for simplicity) the expected outcome is 3.5. If I roll two then I have the following possible outcomes
prob. outcome
1/36 1
3/36 2
5/36 3
7/36 4
9/36 5
11/36 6

for an expected outcome of 4.47

What I'm looking for is a formula to calculate that number given X, the number of dice used, and Y, the number of sides each die has. Hopefully that makes more sense.