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Math Help - Rolling dice probability

  1. #1
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    Rolling dice probability

    Roll an m-sided fair dice n times. What is the probability that all numbers from 1 to k appear at least once and none from k+1 to m appear during the n trials? ( n\geq k)
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  2. #2
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    Re: Rolling dice probability

    Here's my thought. Check if it is correct. I also want to know if there's a simpler method.

    Given any i numbers from 1 to k, let w(i) be the number of scenarios under which all and only those i numbers appear in our n trials. Then we have w(1)=1, and w(i)={i}^{n}-\sum_{j=1}^{i-1}{i \choose j}w(j).

    By this definition, there are exactly {k \choose i}w(i) scenarios under which exactly i different numbers from 1 to k appear.

    Hence the required probability p=({k}^{n}-\sum_{i=1}^{k-1}{k \choose i}w(i))/{m}^{n}.
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