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A ring is hanging from the ceiling by a string. Someone will cut the ring in two positions chosen uniformly at random on the circumference, and this will break the ring into two pieces. Player I gets the piece which falls to the floor, and player II gets the piece which stays attaches to the string. Whoever gets the bigger piece wins. Does either plater have a big advantage here? Explain.

Assuming the circumference has length 1, I need to find the probability that the length of the middle piece exceeds 0.5 but I don't know how to so please help me out.

Thanks

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  • $\begingroup$ "Number theory"?? Not a whiff. $\endgroup$ – Did Feb 5 '14 at 6:18
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Represent the ring by the interval $[0,1]$ with $0$ and $1$ both being the highest point of the ring. The part which falls on the floor is $[x,y]$ with density $$f(x,y)=2\mathbf 1_{0\leqslant x\lt y\leqslant1}, $$ hence this part is the largest one with probability $$ \iint\mathbf 1_{y-x\gt1/2}f(x,y)\mathrm dx\mathrm dy=\int_0^{1/2}\int_{x+1/2}^12\mathrm dy\mathrm dx =\int_0^{1/2}(1-2x)\mathrm dx=\frac14. $$

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