Task 11. Ten different books are randomly arranged on one bookshelf. Find the probability that two specific books will be placed next to each other (event $A$).
Solution. The number $n$ of all elementary outcomes of the experiment in which event $A$ can occur is $C_{10}^{2}$, and the number $m$ of outcomes favorable to event $A$ is 9, since the two specific books will be placed next to each other if they occupy either the first-second positions, the second-third, and so on, up to the ninth-tenth positions on this shelf. Therefore, the probability of event $A$ is:

$$
p(A)=\frac{m}{n}=\frac{9}{C_{10}^{2}}=\frac{9 \cdot 1 \cdot 2}{10 \cdot 9}=0.2
$$

Answer. $p(A)=0.2$.

## Exercises

1. Let $A, B, C$ be three arbitrary events. Find the expressions for the events that will occur in the following cases:
1) only event $A$ occurs;
2) only one event occurs;
3) exactly two events occur;
4) all three events occur;
5) at least one event occurs;
6) no more than two events occur.
2. Two dice are rolled. What is the probability that the absolute value of the difference in the numbers rolled is 2?
3. From $n$ items, of which $r$ are defective, $r$ items are randomly selected. What is the probability that all of them are defective?
4. A student knows 30 out of 40 questions in the syllabus. Find the probability that he knows two out of three questions in his ticket.
5. Inside the circle $x^{2}+y^{2}=a^{2}$, there is an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(b \leqslant a)$. Find the probability of a point landing in the region bounded by the circle and the ellipse.
6. In the first box, there are balls numbered from 1 to 5, and in the second box, there are balls numbered from 6 to 10. One ball is drawn from each box. What is the probability that the sum of the numbers on the drawn balls:
1) is at least 7;
2) is 11;
3) is no more than 11?
7. A deck of 36 cards is randomly divided into two equal halves. Find the probability that each half-deck will be of one color.
8. A group of 17 students occupies a row of the same number of seats in a lecture hall. Find the probability that two specific students will be sitting next to each other.
9. Two dice are rolled. Find the probability that the sum of the numbers rolled is 8.
10. In a lottery, there are 100 tickets, 40 of which are winning. Two tickets are purchased. Find the probability that both are winning.
11. 30 suits out of 50 are made at factory $A$, and the remaining 20 are made at factory $B$. 8 suits are purchased from the 50. Find the probability that 5 of the purchased suits are made at factory $A$.
12. In a batch of 100 items, there are 10 non-standard items. Find the probability that: a) out of 4 items taken, 3 are non-standard; b) out of 5 items, 4 are standard.
13. After a hurricane, telephone communication between points A and B, which are 10 km apart, was disrupted. Find the probability that the break occurred between points C, which is 5 km from A, and D, which is 8 km from A.
14. A parquet floor consists of squares with a side of $20 \, \text{cm}$. A coin with a radius of $1.5 \, \text{cm}$ falls on the floor. Find the probability that it does not cross the boundaries of the squares.
15. Two tourists agreed to meet between 10 am and 11 am. The one who arrives first waits for 15 minutes and, if the second does not come, leaves. Find the probability of the tourists meeting.

## Answers

1. 2) $A \bar{B} \bar{C}$; 2) $A \bar{B} \bar{C}+\bar{A} B \bar{C}+\bar{A} \bar{B} C$; 3) $A B \bar{C}+A \bar{B} C+\bar{A} B C$; 4) $A B C$; 5) $A+B+C$; 6) $\overline{A B C}$. 2. $\frac{8}{36}=0.2222$. 3. $\frac{1}{C_{n}^{r}}$. 4. 0.44.
1. $1-\frac{b}{a}$. 6. 1) 1; 2) 0.2; 3) 0.6. 7. $\frac{2}{C_{36}^{18}} \cdot 8 \cdot \frac{2}{17} \approx 0.118$. 9. $\frac{5}{36}=0.139$. 10. 0.158. 11. $\frac{C_{40}^{2}}{C_{100}^{2}}=\frac{773604}{2556565} \approx 0.303$. 12. a) $\frac{144}{52283} \approx 0.003$; b) $\frac{12905}{38024} \approx 0.339$. 13. 0.3. 14. 0.7225. 15. $\frac{7}{16}=0.1375$.

## § 3. Theorem of Addition of Probabilities of Incompatible Events

Theorem 1. If events $A_{1}, \ldots, A_{n}$ are pairwise incompatible, then the probability of the sum of these events is equal to the sum of their probabilities, i.e.

$$
p\left(A_{1}+A_{2}+\ldots+A_{n}\right)=p\left(A_{\mathrm{I}}\right)+p\left(A_{2}\right)+\ldots+p\left(A_{n}\right)
$$

Corollary 1. The sum of the probabilities of opposite events is equal to one:

$$
p(A)+p(\bar{A})=1
$$

Corollary 2. The sum of the probabilities of events $A_{1}, \ldots, A_{n}$, forming a complete group, is equal to one:

$$
p\left(A_{\mathrm{I}}\right)+p\left(A_{2}\right)+\ldots+p\left(A_{n}\right)=1
$$

## Problems with Solutions