PyPE solution to PE problem 205

    ProjectEuler.net problem 205
    ============================

    Peter has nine four-sided (pyramidal) dice, each with faces numbered 1, 2,
    3, 4. Colin has six six-sided (cubic) dice, each with faces numbered 1, 2,
    3, 4, 5, 6.

    Peter and Colin roll their dice and compare totals: the highest total wins.
    The result is a draw if the totals are equal.

    What is the probability that Pyramidal Pete beats Cubic Colin? Give your
    answer rounded to seven decimal places in the form 0.abcdefg

    Solution
    --------

    From Wikipedia, we can calculate the probability of rolling a certain total
    k using i dice, each with s sides:

        F(s,i,k) =
          (1/s**i) * sum_(n=0..(k-i)/s){ (-1)**n * nCr(i,n) * nCr(k-sn-1,i-1) }

    For any given total k rolled by Peter, the probability of rolling that k is
    given by F(4,9,k). The probability that Colin rolled alower (but not equal)
    total j is given by the sum(F(6,6,j)) for j in 6..(k-1). Therefore, the
    probability that Peter rolled a k and won is the product of these two
    probabilities.

    The total probability that Peter will win a roll is obtained by summing over
    all possible totals k that he could have rolled.

    Answer
    ------

    0.5731441