Determine Frequency of numbers showing up in dice

Question

For a game I'm trying to determine the frequency that a certain # will show up at a given # of dice being rolled. I know... that question seems odd. Let me try to explain it with real numbers.

So, for 1 die, the frequency for each number will be identical. 1-6 will show up equal number of times.

Now for 2 dice, things get different. I imagine 5,6,7 are going to be the most frequently rolled, while numbers at both ends of the spectrum will show up less or not at all (in the case of 1). I'd like to know how to calculate this list and show them in the proper order, from most frequent to less frequent.

Any thoughts?

---

@duffymo - It would be really nice though to have some sort of an algorithm to come up with it. It seems that the above way is going to require a lot of hand picking and placing of numbers. If my die count is dynamic up to say 10, doing that by hand will be ineffecient and troublesome I think. :)

Answer 1

There is no real "algorithm" or simulation necessary - it's a simple calculation based on a formula derived by De Moivre:

http://www.mathpages.com/home/kmath093.htm

And it's not a "bell curve" or normal distribution.

Answer 2

There seems to be some mystery surrounding exactly "why" this is, and although duffymo has explained part of it, I'm looking at another post that says:

There should be no reason why 5, 6 and 7 should be rolled more [than 2] since the first roll of the die is a independent event from the second roll of the die and both of them have equal probablity of 1-6 of being rolled.

There's a certain appeal to this. But it's incorrect...because the first roll affects the chances. The reasoning can probably most easily be done through an example.

Say I'm trying to figure out if the probability of rolling 2 or 7 is more likely on two dice. If I roll the first die and get a 3, what are my chances now of rolling a total of 7? Obviously, 1 in 6. What are my chances of rolling a total of 2? 0 in 6...because there's nothing I can roll on the second die to have my total be 2.

For this reason, 7 is very (the most) likely to be rolled...because no matter what I roll on the first die, I can still reach the correct total by rolling the right number on the second die. 6 and 8 are equally slightly less likely, 5 and 9 more so, and so on, until we reach 2 and 12, equally unlikely at 1 in 36 chance apiece.

If you plot this (sum vs likelyhood) you'll get a nice bell curve (or, more precisely, a blocky aproximation of one because of the discrete nature of your experiment).

Answer 3

There's lots of stuff online about dice probability. Here's one link that helped me out with a Project Euler question:

http://gwydir.demon.co.uk/jo/probability/calcdice.htm