How Do You Calculate the Probability of a Sum in n Dice Rolls?

[Astudious]

How does one calculate the probability of a sum of r in the dice rolls of n dice? Can a probability distribution be written for something like this, to calculate the probability of a sum greater than r, greater than or equal to r, equal to r, less than or equal to r, less than r, etc.?

I do not want to simply brainstorm solutions leading to a sum of r, unless it is easy to generalize from there to n dice (which, from my consideration of n=2 and n=3, doesn't seem like the case - I need a more general way of thinking about it).

 

[Simon Bridge]

How does one calculate the probability of a sum of r in the dice rolls of n dice?

Same way you calculate any probability - the number of ways it can happen divided by the total number of things that can happen.

Can a probability distribution be written for something like this, to calculate the probability of a sum greater than r, greater than or equal to r, equal to r, less than or equal to r, less than r, etc.?

Yes.

I do not want to simply brainstorm solutions leading to a sum of r, unless it is easy to generalize from there to n dice (which, from my consideration of n=2 and n=3, doesn't seem like the case - I need a more general way of thinking about it).

The more general way is above, and going through specifics does generalize eventually - it's just not so obvious. Perhaps revisit combinatorics?

 

[Astudious]

Same way you calculate any probability - the number of ways it can happen divided by the total number of things that can happen.

Yes.

The more general way is above, and going through specifics does generalize eventually - it's just not so obvious. Perhaps revisit combinatorics?

What should I use to revisit combinatorics properly? I've read the Schaum's Outline but it doesn't seem to cover problems of this standard.

 

[Simon Bridge]

Schaum probably has enough information to get you going - you only need the basic concepts: the bit before "combination" and "permutation" notation is defined.
OR you can just http://www.mathpages.com/home/kmath093.htm and see how others have done it...

 

[Stephen Tashi]

There is a general method related to the combinatorics of the problem. Consider the product:

$(x + x^2 + x^3 + x^4 + x^5 + x^6)^3$

If you multiply it out and combine like terms then, for example, the coefficient of x^4 gives the number of sequences of 3 die rolls that produces a sum of 4.

This method is an example of using "generating functions" in combinatorics The term "generating function" has different meanings in different branches of math, so to search for the topic on the web, you should use a more specific search that the keywords "generating function". There are many examples on the web of using this type of generating function as a step in solving problems like "How many ways can you pay a dollar debt if you have 3 quarters, 5 dimes, and 6 nickels".