Clement's conjecture

comparison.jl

using Distributions

# Search counter examples in exact arithmetic

xpdf(d::Binomial,k) = binomial(d.n, k)*d.p^k*(1 - d.p)^(d.n - k)

f(x,y) = x*y + (1-x)*(1-y)
   
while true

    p = big(rand(1:250)//rand(500:1000))
    q = big(rand(1:250)//rand(500:1000))
    p, q = minmax(p, q)

    m, n = rand(2:10), rand(2:10)
    r =  big(rand(1:250)//rand(500:1000))
    a = sum((f(i//m, j//n)>r)*xpdf(Binomial(m, p), i)*xpdf(Binomial(n, p), j) for i in 0:m, j in 0:n)
    b = sum((f(i//m, j//n)>r)*xpdf(Binomial(m, q), i)*xpdf(Binomial(n, q), j) for i in 0:m, j in 0:n)
    if a < b 
        @show n, m, r, p, q
    end
end