Problem 339
Peredur fab Efrawg
“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.”
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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.
埃夫罗格之子佩雷迪尔
“他走向一座山谷，山谷中有河流经过；山谷的四周被树木环绕，河流的两岸有平整的牧草。在河的一边他看到一群白色的绵羊，在河的另一边他看到一群黑色的绵羊。每当一只白色绵羊叫唤时，一只黑色绵羊会穿过河流并变成白色；每当一只黑色绵羊叫唤时，一只白色绵羊会穿过河流变成黑色。
译自英文维基百科
一开始，两群羊均为n只。每只羊（无论颜色）都等概率地成为下一只会叫的羊。在有一只羊叫唤，另一只羊穿过河流变色之后，佩雷迪尔可以移除一定数量的白色绵羊，以最大化黑色绵羊最终数目的期望值。记E(n)是在佩雷迪尔使用最优策略时黑色绵羊最终数目的期望值。
已知保留小数点后6位小数时E(5) = 6.871346。
求E(10 000)，并将你的答案保留小数点后6位小数。