695. Random Rectangles

Three points, P1, P2 and P3, 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 P1P2, P1P3 or P2P3 (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 P2P3.

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 P1P2P3。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 P1P2P1P3P2P3 三者之一。(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 P2P3 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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