695. 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.

695. 随机矩形

在单位正方形上，随机选取三个点 $P_1$、$P_2$ 和 $P_3$。考虑三个满足如下条件的矩形：其有与单位正方形平行的边，且其对角线为 $\overline{P_1P_2}$，$\overline{P_1P_3}$，$\overline{P_2P_3}$ 三条线段中的一条。（见下图）

我们对于面积第二大的矩形很感兴趣。本例中，面积第二大的是以 $\overline{P_2P_3}$ 为对角线的绿色矩形。

求面积第二大的矩形的期望面积，并将你的答案四舍五入至小数点后第 10 位。

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