[ Average values

Vasya, in a rage, cuts a rectangular sheet of paper with scissors. Every second, he cuts the first piece he comes across with a random straight cut into two parts.

a) Find the expected number of sides of the polygon that will randomly fall to Vasya after an hour of such work

b) Solve the same problem if the sheet of paper initially had the shape of an arbitrary polygon.

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b) Let as a result of Vasya's activity, there are $k+1$ polygons on the table, including $p_{3}$ triangles, $p_{4}$ quadrilaterals, $\ldots, p_{m} m$-gons ($m$ - the largest possible number of sides of the resulting polygon). Then the expected number of sides of the selected polygon is $1 / k+1\left(3 p_{3}+4 p_{4}+\ldots+m p_{m}\right)$.

In parentheses is the total number of sides of all the obtained polygons.

Suppose Vasya had an $n$-gon. The probability of making a random cut exactly through some vertex is zero. Therefore, each cut divides some two sides of the polygon being cut into two parts each, and two new sides are formed along the cut. Thus, each cut adds 4 new sides. After $k$ cuts, the total number of sides will be $n+4 k$.

Thus,

a) If Vasya initially had a quadrilateral, then $\mu=\frac{4+4 k}{k+1}=4$.

## Answer

a) 4; b) $\frac{n+14400}{3601}$, where $n$ is the number of sides of the initial polygon.