83 lines
5.4 KiB
HTML
83 lines
5.4 KiB
HTML
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<h2>Peredur fab Efrawg</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 339</h3><span style="width:300px;color:#666;">Published on Sunday, 22nd May 2011, 04:00 am; Solved by 337</span></div>
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<i>"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."</i><BR />
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<a href=http://en.wikisource.org/wiki/The_Mabinogion/Peredur_the_Son_of_Evrawc>en.wikisource.org</A>
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Initially each flock consists of <var>n</var> 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(<var>n</var>) be the expected final number of black sheep if Peredur uses an optimal strategy.
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<P>
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You are given that E(5) = 6.871346 rounded to 6 places behind the decimal point.<BR />
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Find E(10 000) and give your answer rounded to 6 places behind the decimal point.
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