"And he came towards a valley, through which ran a river; and the
borders of the valley were wooded, and on each side of the river were
level meadows. And on one side of the river he saw a flock of white
sheep, and on the other a flock of black sheep. And whenever one of the
white sheep bleated, one of the black sheep would cross over and become
white; and when one of the black sheep bleated, one of the white sheep
would cross over and become black."
Initially each flock consists of n sheep. Each sheep (regardless of colour) is equally likely to be the next sheep to bleat. After a sheep has bleated and a sheep from the other flock has crossed over, Peredur may remove a number of white sheep in order to maximize the expected final number of black sheep. Let E(n) be the expected final number of black sheep if Peredur uses an optimal strategy.
You are given that E(5) = 6.871346 rounded to 6 places behind the decimal point.
Find E(10 000) and give your answer rounded to 6 places behind the decimal point.