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ProjectEuler/pe/p285.html
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<!DOCTYPE html>
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<title>Problem 285 - Project Euler</title>
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<h2>Pythagorean odds</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 285</h3><span style="width:300px;color:#666;">Published on Saturday, 3rd April 2010, 05:00 am; Solved by 685</span></div>
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<div class="problem_content" role="problem">
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<p>Albert chooses a positive integer <var>k</var>, then two real numbers <var>a</var>, <var>b</var> are randomly chosen in the interval [0,1] with uniform distribution.<br />
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The square root of the sum (<var>k</var>·<var>a</var>+1)<sup>2</sup> + (<var>k</var>·<var>b</var>+1)<sup>2</sup> is then computed and rounded to the nearest integer. If the result is equal to <var>k</var>, he scores <var>k</var> points; otherwise he scores nothing.</p>
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<p>For example, if <var>k</var> = 6, <var>a</var> = 0.2 and <var>b</var> = 0.85, then (<var>k</var>·<var>a</var>+1)<sup>2</sup> + (<var>k</var>·<var>b</var>+1)<sup>2</sup> = 42.05.<br />
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The square root of 42.05 is 6.484... and when rounded to the nearest integer, it becomes 6.<br />
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This is equal to <var>k</var>, so he scores 6 points.</p>
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<p>It can be shown that if he plays 10 turns with <var>k</var> = 1, <var>k</var> = 2, ..., <var>k</var> = 10, the expected value of his total score, rounded to five decimal places, is 10.20914.</p>
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<p>If he plays 10<sup>5</sup> turns with <var>k</var> = 1, <var>k</var> = 2, <var>k</var> = 3, ..., <var>k</var> = 10<sup>5</sup>, what is the expected value of his total score, rounded to five decimal places?</p>
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