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Problem 695

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Random Rectangles

Three points, $P_1$, $P_2$ and $P_3$, are randomly selected within a unit square. Consider the three rectangles with sides parallel to the sides of the unit square and a diagonal that is one of the three line segments $\overline{P_1P_2}$, $\overline{P_1P_3}$ or $\overline{P_2P_3}$ (see picture below).

We are interested in the rectangle with the second biggest area. In the example above that happens to be the green rectangle defined with the diagonal $\overline{P_2P_3}$.

Find the expected value of the area of the second biggest of the three rectangles. Give your answer rounded to $10$ digits after the decimal point.

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随机长方形

在单位正方形的边缘随机选择三个点$P_1$、$P_2$和$P_3$，并分别以线段$\overline{P_1P_2}$、$\overline{P_1P_3}$和$\overline{P_2P_3}$为对角线作各边平行于单位正方形的三个长方形，如下图所示：

我们感兴趣的是这其中面积第二大的那个长方形。在上述例子当中，面积第二大的长方形是以线段$\overline{P_2P_3}$为对角线的绿色长方形。

求面积第二大的长方形的面积期望值，并将你的答案保留小数点后$10$位数字。

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