# co.combinatorics – A ratio of two probabilities

Ley $$t:=eta$$. Then
$$f(t)=frac{P(Gge K)}{P(B ge K)},$$
where $$G$$ is a random variable with the binomial distribution with parameters $$N,q_Gt$$ and $$B$$ is a random variable with the binomial distribution with parameters $$N,q_Bt$$; here we must assume that $$q_B>0$$ and $$tin(0,1/g_G)$$, so that $$q_Gt$$ and $$q_Bt$$ are in the interval $$(0,1)$$.

The random variables $$G$$ and $$B$$ have a monotone likelihood ratio (MLR): for each $$xin{0,dots,N}$$,
$$frac{P(G=x)}{P(B=x)}=CBig(frac{1-q_Gt}{1-q_Bt}Big)^{N-x},$$
which is decreasing in $$tin(0,1/g_G)$$; here, $$C$$ is a positive real number which does not depend on $$t$$.

It is well known that the MLR implies the MTR, the monotone tail ratio. Thus, the desired result follows.