Example 12. The region $G$ is bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, and the region $g$ - by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly fixed in the region $G$. What is the probability that it belongs to the region $g$ (event $A$)?
Solution. The volume $V$ of the body bounded by the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ is given by the formula

$$
V=\frac{4}{3} \pi a b c
$$

therefore

$$
V_{\mathrm{G}}=\frac{4}{3} \pi 4 \cdot 3 \cdot 2=32 \pi
$$

The volume of a sphere of radius $R$ is given by the formula

$$
V=\frac{4}{3} \pi R^{3},
$$

so for $R=2$ we have

$$
V_{1}=\frac{4}{3} \pi \cdot 2^{3}=\frac{32}{3} \pi
$$

The volume of the region $g$:

$$
V_{G}-V_{1}=32 \pi-\frac{32}{3} \pi=\frac{96-32}{3} \pi=\frac{64}{3} \pi, V_{G}=\frac{64}{3} \pi
$$

Using formula (1.5.2), we calculate the desired probability:

$$
P(A)=\frac{V_{g}}{V_{G}}=\frac{2}{3}
$$

## Problems

1. On a plane, two concentric circles are drawn with radii of 6 and $12 \mathrm{~cm}$ respectively. What is the probability that a point thrown randomly into the larger circle will fall into the ring formed by the specified circles?
2. A regular triangle is inscribed in a circle of radius $R$. Find the probability that a point thrown into this circle will fall into the given triangle.
3. A point $Q(x, y)$ is thrown randomly into a square with vertices $O(0,0), K(0,1), L(1,1), M(1,0)$. What is the probability that the coordinates of this point satisfy the inequality $y>2 x ?$
4. A regular triangular pyramid is inscribed in a sphere. A point is randomly fixed in the sphere. Find the probability that the point will fall into the pyramid.
5. A rod of length $l$ is broken into three parts at random. What is the probability that a triangle can be formed from these parts?
6. On a plane, the region $G$ is bounded by the ellipse $x^{2} / 36+y^{2} / 25=1$, and the region $g$ by this ellipse and the ellipse $x^{2} / 9+y^{2} / 4=1$. A point is thrown into the region $G$. What is the probability that the point will fall into the region $g$?
7. A point is thrown into a rectangle with vertices $K(-2,0), L(-2,5), \quad M(1,5)$, $N(1,0)$. What is the probability that its coordinates $(x, y)$ will satisfy the inequalities $x^{2}+1 \leq y \leq 3-x$?
8. In the region $G$, bounded by the ellipsoid $\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{4}=1$, a point is randomly fixed. What is the probability that the coordinates $x$, $y, z$ of this point will satisfy the inequality $x^{2}+y^{2}+z^{2} \leq 4$?
9. A point is thrown into a rectangle with vertices $R(-2,0), L(-2,9), M(4,9), N(4,0)$. Find the probability that its coordinates will satisfy the inequalities $0 \leq y \leq 2 x-x^{2}+8$.
10. The region $G$ is bounded by the circle $x^{2}+y^{2}=25$, and the region $g$ by this circle and the parabola $16 x-3 y^{2}=0$. A point is thrown into the region $G$. What is the probability that it will fall into the region $g$?

## Answers

1. 0.75. 2. $3 \sqrt{3} / 4 \pi \approx 0.4137$. Hint. The side $a$ of the triangle in terms of the radius $R$ of the circumscribed circle is given by the formula $a=\sqrt{3} R$. 3. 0.25. 4. $2 / 3 \sqrt{3} \pi \approx 0.123$. Hint. $a=4 R / \sqrt{6}, 5 \cdot 0.25, 6.5 / 6.7 \cdot 0.3.8 \cdot 1 / 3.9. 2 / 3$. 10. $\approx 0.352$.

## Questions

1. How is geometric probability defined in general?
2. How is geometric probability defined in the spatial case?
3. How is geometric probability defined in the planar case?
4. How is geometric probability defined in the linear case?
5. What are the properties of geometric probability?
6. Provide your own example of geometric probability.

## § 1.6. Operations on Events. Relations Between Events

The sum, or union, of two events is the event that consists of the occurrence of at least one of them. The sum of two events $A$ and $B$ is denoted by $A+B$ or $A \cup B$. Similarly, the sum of $n$ events is defined and denoted as the event that consists of the occurrence of at least one of them. The sum of $n$ events $A_{1}, A_{2}, \ldots, A_{n}$ is denoted as follows:

$$
A_{1}+A_{2}+\ldots+A_{n}=\sum_{k=1}^{n} A_{k}, \text { or } A_{1} \cup A_{2} \cup \ldots \cup A_{n}=\cup_{k=1}^{n} A_{k}
$$

The product, or intersection, of two events is the event that consists of their simultaneous occurrence. The product of two events $A$ and $B$ is denoted by $A B$ or $A \cap B$. Similarly, the product in the case of a larger number of events is defined and denoted. The product of $n$ events $A_{1}, A_{2}, \ldots, A_{n}$ is denoted as follows:

$$
A_{1} A_{2} \ldots A_{n} \text { or } A_{1} \cap A_{2} \cap \ldots \cap A_{n}
$$

The concepts of the sum and product of events are extended to infinite sequences of events. In these cases, for example, the following notations are used:

$$
\begin{aligned}
& A_{1} \cup A_{2} \cup \ldots \cup A_{n} \cup \ldots=\bigcup_{k=1}^{\infty} A_{k} \\
& A_{1} \cap A_{2} \cap \ldots \cap A_{n} \cap \ldots=\bigcap_{k=1}^{\infty} A_{k}
\end{aligned}
$$

If event $A$ will necessarily occur when some other event $B$ occurs, then it is said that event $B$ is a particular case of event $A$, and we write $B \subset A$, or $A \supset B$ (it is also said that $B$ implies $A$).

If $B \subset A$ and $A \subset B$, i.e., events $A$ and $B$ can occur or not occur together in the given experiment, then they are called equivalent, and we write $A=B$.

Operations of union and intersection of events have some properties analogous to the properties of addition and multiplication of numbers. These operations are commutative:

$$
A \cup B=B \cup A, A \cap B=B \cap A \text {; }
$$

associative:

$$
\begin{aligned}
& (A \cup B) \cup C=A \cup(B \cup C)=(A \cup C) \cup B=A \cup B \cup C \\
& (A \cap B) \cap C=A \cap(B \cap C)=(A \cap C) \cap B=A \cap B \cap C
\end{aligned}
$$

and distributive:

$$
(A \cup B) \cap C=(A \cap C) \cup(B \cap C)
$$

The properties mentioned follow from the definitions of the operations of union and intersection of events.

Not all laws of addition and multiplication of numbers are valid for the union and intersection of events. For example, for any event $A$ the following equalities hold:

$$
A \cup A=A, A \cap A=A \text {. }
$$

If $U$ is a certain, $V$ is an impossible event, $A$ is any event, $\bar{A}$ is the event opposite to $A$, then the following equalities hold:

$$
\begin{aligned}
& A \cap \bar{A}=V, \text { or } A \cdot \bar{A}=V . \\
& A \cup \bar{A}=U, \text { or } A+\bar{A}=U .
\end{aligned}
$$

$$
\begin{aligned}
& A \cup V=A, \text { or } A+V=A \\
& A \cap V=V, \text { or } A \cdot V=V \\
& A \cup U=U, \