Hi, Dear All,

I come up with this problem, which I think for a long time without a good answer.

Suppose two independent random variables $X \sim \mathrm{Binomial}(n, p)$ and $Y \sim \mathrm{Binomial}(m, p)$, and n < m .

We know that: \begin{eqnarray} Pr(X>Y) = \sum_{k_1=1}^n\sum_{k_2=1}^{m}\mathrm{I}(k_1>k_2) {n \choose k_1}{m \choose k_2}p^{k_1+k_2}(1-p)^{m+n-k_1-k_2} \end{eqnarray} \begin{eqnarray} = \sum_{k_1=1}^n\sum_{k_2=1}^{k_1 -1}{n \choose k_1}{m \choose k_2}p^{k_1+k_2}(1-p)^{m+n-k_1-k_2} \end{eqnarray}

**Question:**

Do we have closed form for the above problem?

Motivation: because I want to solve Pr(X>Y| n< m) > a, and try to bound the p, so I have to get a closed form and try to solve this inequality.

$$\sum_{k_1=1}^n\sum_{k_2=1}^{k_1 -1}{n \choose k_1}{m \choose k_2}p^{k_1+k_2}(1-p)^{m+n-k_1-k_2} > a$$

$$ p > ???$$

Any hints? Thanks.