Example. The plane is divided into square cells with side $a$ by straight lines. The cells are numbered using pairs of integers $(m, n)$ (thus, the cell to the right will have the number $(m+1, n)$, and the cell above will have the number $(m, n+1)$). Find the probability that a coin of radius $r<a / 2$ thrown randomly on the plane will completely fall into a cell for which $m+n$ is divisible by three.
Solution. Randomly tossing a coin is equivalent to randomly selecting a point where the center of the coin will be located. The desired probability is the ratio of the area of the favorable outcomes region in one cell to the area of the entire cell.

1. Define the sample space.

In this case, it consists of equal rectangular cells into which the plane is divided and which contain equal parts of the set of favorable outcomes $G$ and equal parts of its complement to the entire plane. Denote such a cell by the symbol $D$. To find $D$, note that if a certain cell $(m, n)$ satisfies the required condition, then the cells $(m+3, n),(m-3, n),(m, n+3),(m, n-3)$ also satisfy it, i.e., there is periodicity horizontally with a period of 3 and vertically with a period of 3. Therefore, the cell $D$ can be taken as the region composed of the cells $(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1)$, and $(2,2)$.

2. Check the equiprobability of the elementary outcomes.

In this case, it is guaranteed by the methodology of conducting the random experiment, as stated in the problem, the coin is tossed randomly.

3. Select the elementary outcomes that lead to the occurrence of the event of interest (favorable outcomes).

Clearly, among the cells of the cell $D$, the sum of the coordinates $m+n$ is divisible by three only for the cells $(0,0),(1,2)$, and $(2,1)$.

For the coin to be entirely within a certain cell, it is necessary and sufficient that the center of the coin is located at one of the points inside the cell at a distance greater than $r$ from the boundary of this cell. In Fig. 6.2, the cell $D$ is depicted and the region of favorable outcomes $G \cap D$ is shaded.

![](https://cdn.mathpix.com/cropped/2024_05_22_718fd8b5b0aaabbbc775g-265.jpg?height=628&width=803&top_left_y=1155&top_left_x=331)

Fig. 6.2

4. Find the area $S(D)$ of the cell $D$. We have $S(D)=9 a^{2}$.
5. Find the area $S(G \cap D)$ of the region $G \cap D$. We have $S(G \cap D)=$ $=3(a-2 r)^{2}$.
6. The desired probability is

$$
\frac{S(G \cap D)}{S(D)}=\frac{(a-2 r)^{2}}{3 a^{2}}
$$

Answer. $(a-2 r)^{2} /\left(3 a^{2}\right)$.

Problem Statement. The plane is divided by lines into square cells with side $a$. The cells are numbered using pairs of integers $(m, n)$ (in this case, the cell to the right will have the number $(m+1, n)$, and the cell above will have the number $(m, n+1))$. A coin of radius $r<a / 2$ is randomly tossed onto the plane. Find the probability that the coin will completely fall into a cell with a number satisfying the specified condition.

1. $m$ is even.
2. $m+n$ is even.
3. $m^{2}+n^{2}$ is divisible by three.
4. Either $m$ is even or $n$ is divisible by three.
5. $m$ is divisible by three, and $n$ is not divisible by three.

6. Neither $m$ nor $n$ is divisible by three.
7. $m+n$ is divisible by four.

8. Either $m$ is divisible by three or $n$ is divisible by three.
9. $m+2 n$ is divisible by three.
10. $m^{2}-n^{2}$ is divisible by four.

Answers. 1. $(a-2 r)^{2} /\left(2 a^{2}\right)$. 2. $(a-2 r)^{2} /\left(2 a^{2}\right)$. 3. $(a-2 r)^{2} /\left(9 a^{2}\right)$. 4. $2(a-2 r)^{2} /\left(3 a^{2}\right)$. 5. $2(a-2 r)^{2} /\left(9 a^{2}\right)$. 6. $4(a-2 r)^{2} /\left(9 a^{2}\right)$. 7. $(a-2 r)^{2} /\left(4 a^{2}\right)$.
8. $5(a-2 r)^{2} /\left(9 a^{2}\right)$.
9. $(a-2 r)^{2} /\left(3 a^{2}\right)$. 10. $(a-2 r)^{2} /\left(2 a^{2}\right)$.

## 6.6. Independent Events

Problem Statement. Suppose a certain random event $A$ can be expressed in terms of independent random events $A_{1}, A_{2}, \ldots, A_{n}$ using operations of union, intersection, and subtraction. The probabilities of events $A_{1}, A_{2}, \ldots, A_{n}$ are known. The task is to find the probability of event $A$.

## Solution Plan.

1. Express the event $A$ in terms of $A_{1}, A_{2}, \ldots, A_{n}$.
2. Calculate the probability of the event $A$ using the following formulas for independent random events $B_{1}, B_{2}, \ldots, B_{m}$:

a) $\mathrm{P}\left(B_{1} \cap B_{2} \cap \ldots \cap B_{m}\right)=\prod_{i=1}^{m} \mathrm{P}\left(B_{i}\right)$;

b) $\mathrm{P}\left(B_{1} \cup B_{2} \cup \ldots \cup B_{m}\right)=1-\prod_{i=1}^{m}\left(1-\mathrm{P}\left(B_{i}\right)\right)$.