## Range flips

### Problem 430

Published on 02 June 2013 at 07:00 am [Server Time]

`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(10^{10}, 4000).

Give your answer rounded to 2 decimal places behind the decimal point.

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