Example 10. Two positive numbers, each not greater than one, are taken at random. What is the probability that their sum does not exceed one and their product is no more than $2 / 9?$
Solution. Let the taken numbers be denoted by $x$ and $y$. Their possible values satisfy the inequalities: $0 \leq x \leq 1, 0 \leq y \leq 1$, which on the plane define a unit square with area $S_{G}=1$. Favorable values of $x$ and $y$ are determined by the conditions: $x+y \leq 1$, $x \cdot y \leq 2 / 9$; the region $g$ is bounded by segments of the line $x+y=1$ and the arc of the hyperbola $x \cdot y=2 / 9$. The abscissas of the points of intersection of the line and the hyperbola are: $\quad x_{1}=1 / 3, \quad x_{2}=2 / 3$. The line $x+y=1$ divides the square in half, and the region $x+y \leq 1$ represents the lower triangle (Fig. 1.14). The region $g$ consists of a trapezoid, a curvilinear trapezoid, and a triangle, its area

$$
S_{g}=S_{1}+S_{2}+S_{3}
$$

where

$$
S_{1}=5 / 18, S_{3}=1 / 18
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_720b3cbdc89c57e2b629g-035.jpg?height=468&width=483&top_left_y=39&top_left_x=608)

Fig. 1.14

$$
\begin{gathered}
S_{2}=\frac{2}{9} \int_{1 / 3}^{2 / 3} \frac{d x}{x}=\frac{2}{9} \ln 2 \approx \frac{2}{9} \cdot 0.6931 \approx 0.154 \\
S_{g}=\frac{1}{3}+0.154 \approx 0.467
\end{gathered}
$$

Therefore, the desired probability is

$$
P=\frac{S_{g}}{S_{G}}=0.467
$$