Problem 430
Range flips
N disks are placed in a row, indexed 1 to N from left to right.
Each disk has a black side and white side. Initially all disks show their white side.
At each turn, two, not necessarily distinct, integers A and B between 1 and N (inclusive) are chosen uniformly at random.
All disks with an index from A to B (inclusive) are flipped.
The following example shows the case N = 8. At the first turn A = 5 and B = 2, and at the second turn A = 4 and B = 6.
Let E(N, M) be the expected number of disks that show their white side after M turns.
We can verify that E(3, 1) = 10/9, E(3, 2) = 5/3, E(10, 4) ≈ 5.157 and E(100, 10) ≈ 51.893.
Find E(1010, 4000).
Give your answer rounded to 2 decimal places behind the decimal point.
区域翻转
N个盘子放置成一排，从左至右分别标记为1至N。
每个盘子都是一面黑一面白，初始时所有的盘子都是白面朝上。
每一轮,从1至N（包含）中随机等概率选出两个数A和B，这两个数可以相同。
所有标记从A至B（包含）的盘子都要翻转。
下面的例子是N = 8时可能的情景。第一轮A = 5而B = 2，第二轮A = 4而B = 6。
记E(N, M)是在M轮后白面向上的盘子数目的期望值。
可以验证，E(3, 1) = 10/9，E(3, 2) = 5/3，E(10, 4) ≈ 5.157以及E(100, 10) ≈ 51.893。
求E(1010, 4000)。
保留小数点后2位小数。