## Task A-4.5.

What is the probability that for two randomly chosen numbers $x, y$ from the set $[-2,2]$ the following holds

$$
|x|+|y| \geqslant 1 \quad \text { and } \quad \| x|-| y|| \leqslant 1 ?
$$
## First Solution.

Let $\Omega=\{(x, y) \in[-2,2] \times[-2,2]:|x|+|y| \geqslant 1,|| x|-| y|| \leqslant 1\}$. The probability that a point $(x, y)$ lies in $\Omega$ is the ratio of the area of the set $\Omega$ to the area of the set $[-2,2] \times[-2,2]$. The second set is a square with side length 4. To determine the area of the set $\Omega$, we will draw it in the plane, considering each inequality separately.

By examining each quadrant, we find that the inequality $|x|+|y| \geqslant 1$ is satisfied by points that lie outside the interior of a square with vertices at $(1,0),(0,1),(-1,0)$, and $(0,-1)$.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-22.jpg?height=631&width=626&top_left_y=975&top_left_x=635)

The inequality $||x|-|y|| \leqslant 1$ is equivalent to $-1 \leqslant|x|-|y| \leqslant 1$. By examining the first and third quadrants separately, and then the second and fourth quadrants, we obtain the region shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-22.jpg?height=637&width=628&top_left_y=1800&top_left_x=637)

Taking the intersection of the obtained sets with the square $[-2,2] \times[-2,2]$, we get the shaded region in the plane as shown in the following figure.

![](https://cdn.mathpix.com/cropped/2024_05_30_4cdaff2107ef283b1d94g-23.jpg?height=645&width=626&top_left_y=177&top_left_x=635)

We will calculate the area of the shaded region (set $\Omega$) as four times the area of the shaded region in one quadrant. In each quadrant, one square with side length $\sqrt{2}$ and one isosceles right triangle with legs of length 1 are shaded. Therefore, the area of the set $\Omega$ is

$$
4 \cdot\left(\sqrt{2}^{2}+\frac{1 \cdot 1}{2}\right)=10
$$

The area of the square $[-2,2] \times[-2,2]$ is 16, so the desired probability is $\frac{10}{16}=\frac{5}{8}$.

Note: The area of the set $\Omega$ can be calculated in multiple ways, for example, as the sum of the areas of eight trapezoids obtained by drawing the diagonals of the large square or by subtracting the areas of the unshaded regions from the area of the large square.