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In $1 \times 1 $ randomly select a point. Let $x > 0$. Find the probability of:

  1. The distance between the point and fixed side of the square is not more than $x$.
  2. The distance between the point and closest side of the square is not more than $x$.
  3. The distance between the point and diagonals is not more than $x$.

As I understand, the answer for the first question is the relation of area of $1 \times x$ rectangle to the area of square. That is, $\frac{1 \cdot x}{1} = x$. I can't understand what changes in the second question though and where to start in the third question.

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1 Answer 1

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In the first question, the distance between the point and fixed side is not more than $x$ in the rectangle $1\times x$ near this fixed side. And in the second question, the distance is not more than $x$ in every $1\times x$ rectangle adjacent to the side. So, the probability is $\frac{1-(1-2x)^2}{1}$ (Draw it and you'll see).

In the third, you should calculate the area of two $x$-wide layers adjacent to the diagonal on both sides (it obviously should not go beyond your $1\times 1$ square).

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  • $\begingroup$ Can you check the answer for the second question please? Mine is $\frac{1-(1-2x)^2}{1}$. Is that correct? $\endgroup$ Oct 7, 2019 at 23:33
  • $\begingroup$ Yes, it's my mistake. $\endgroup$
    – Kubrick
    Oct 7, 2019 at 23:37

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