Problem 783

Urns

Given $n$ and $k$ two positive integers we begin with an urn that contains $k n$ white balls. We then proceed through $n$ turns where on each turn $k$ black balls are added to the urn and then $2 k$ random balls are removed from the urn.

We let $B_{t} (n, k)$ be the number of black balls that are removed on turn $t$ .

Further define $E (n, k)$ as the expectation of $\sum_{t = 1}^{n} B_{t} (n, k)^{2}$ .

You are given $E (2, 2) = 9.6$

Find $E (10^{6}, 10)$ . Round your answer to the nearest whole number.

罐子

对于给定的正整数 $n$ 和 $k$ ，考虑一个一开始装有 $k n$ 个白球的罐子。在之后的 $n$ 轮中，每一轮我们向罐中加入 $k$ 个黑球，再随机拿出 $2 k$ 个任意颜色的球，

记 $B_{t} (n, k)$ 为在第 $t$ 轮拿出的黑球的数目。

再记 $E (n, k)$ 为 $\sum_{t = 1}^{n} B_{t} (n, k)^{2}$ 的期望值。

已知 $E (2, 2) = 9.6$ 。

求 $E (10^{6}, 10)$ ，并将你的答案四舍五入至整数。