require "table2"

local function throw_die(nsides, ndice, s, counts)
    if ndice == 0 then
        counts[s + 1] += 1
        return
    end
    for i = 1, nsides do throw_die(nsides, ndice - 1, s + i, counts) end
end

local function beating_probability(nsides1, ndice1, nsides2, ndice2)
    local len1 = (nsides1 + 1) * ndice1
    local c1 = table.rep(len1, 0)
    throw_die(nsides1, ndice1, 0, c1)

    local len2 = (nsides2 + 1) * ndice2
    local c2 = table.rep(len2, 0)
    throw_die(nsides2, ndice2, 0, c2)

    local p12 = (nsides1 ^ ndice1) * (nsides2 ^ ndice2)
    local tot = 0
    for i = 0, len1 - 1 do
        for j = 0, math.min(i, len2) - 1 do
            tot += c1[i + 1] * c2[j + 1] / p12
        end
    end
    return tot
end

print(beating_probability(4, 9, 6, 6))
print(beating_probability(10, 5, 7, 6))