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I have basic understanding of probability but I couldn't get following question. Please help me to get it.

There are $n$ dice each with $m$ faces numbered from $1$ to $m$. Given a number $x$, find out the probability that the sum of all numbers obtained from the $n$ dice in a trial is greater than $x$.

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  • Fact: Probability generating function (pgf) for the score of one die is $$\mathcal{P}(z) = \frac{1}{m} \left(z+z^2+\cdots+z^{m}\right) = \frac{z}{m} \frac{1-z^{m}}{1-z}$$
  • Fact: The pgf of the total score (assuming independence of scores) $X$ is $\left(\mathcal{P}(z)\right)^n$.
  • Fact: The generating function for the sequence $\mathbb{P}(X>k)$ is $$ \sum_{k=0}^{n \cdot m} z^k \mathbb{P}(X>k) = \frac{1- \left(\mathcal{P}(z)\right)^n}{1-z}$$

Thus the answer to your question is $$ \mathbb{P}(X>k) = [z^k] \frac{1-\left(\frac{z}{m} \frac{1-z^{m}}{1-z}\right)^n}{1-z} $$

A good reference on the subject is the book "Problems and Snapshots from the World of Probability" by Blom, Holst and Sandell. See section 4.6 on page 39.


Here is numerical confirmation in Mathematica:

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