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Sum of n elements of a finite set of integers, 1 through s

  1. Jun 11, 2010 #1
    The general problem I'm trying to solve is the probability of rolling a total t on n s-sided dice. A good chunk of the problem is easy enough, but where I run into difficulty is this:

    How many combinations of dice will yield a sum total of t? Because the number set is limited, [tex]{a \choose n-1}[/tex] (where [tex]a={n(s+1) \over 2} - \left|{n(s+1) \over 2} - t\right|[/tex]) no longer works when [tex]n+s \leq t \leq (n-1)s[/tex]. It is this region in the middle that interests me. Enumerating all combinations could be time-consuming, and, I expect, is entirely unnecessary. Is there a known formula for computing these numbers?
     
    Last edited: Jun 11, 2010
  2. jcsd
  3. Jun 11, 2010 #2
    sum of the m rolls diverges as m gets large. but we can use standardize to make it converge to standard normal distribution. this the weak law of large number says.
     
  4. Jun 11, 2010 #3
    I'm not sure I understand what you're saying. The total t must be in the range [n, ns]. I am aware of the idea that it would probably converge to a normal distribution, but for [tex]{1 \over {\sqrt{2\pi\sigma^2}}} e^{-{{\left((x-{n(s+1) \over 2})-\mu\right)^2} \over {2\sigma^2}}}[/tex], what would [tex]\sigma[/tex] and [tex]\mu[/tex] be in terms of n, s and t?
     
  5. Jun 12, 2010 #4
  6. Jun 12, 2010 #5
    That's exactly what I was looking for, thanks!
     
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