丘成桐数学竞赛原题

hawkzmy

A box contains 750 red balls and 250 blue balls. Repeatedly pick a ball
uniformly at random from the box and remove it until all remaining balls have a single
color. (Note: no replacement).
Please find integer m such that the expectation value for the total number of the
remaining balls ∈ [m, m + 1]
一个盒子里有750个红球和250个蓝球。反复地从盒子里均匀地挑选一个球，然后去掉它，直到所有剩下的球都有一个颜色。（注：没有更换）。请找到整数m，使剩余球总数的期望值为∈[m，m + 1]

yigo

不知道理解的对不，设i个红球，j个篮球，期望为E(i,j)，E(1,1)=1。

\(\displaystyle E(1,i)=E(i,1)=\frac{1}{1+i}i+\frac{i}{1+i}E(1,i-1)=\frac{i^2+i+2}{2(1+i)}\)

\(\displaystyle E(i,j)=\frac{i}{i+j}E(i-1,j)+\frac{j}{i+j}E(i,j-1)\)

excel拉了下，E(750,250)=3.32，故题中的m=3。

归纳了下，通项公式为：

\(\displaystyle E(i,j)=\frac{i}{j+1}+\frac{j}{i+1}\)

hawkzmy

750/251+250/751=3.320937.......

lihpb00

最小拿250次，最多拿750次，750/251+250/751