Example 14. A die is rolled 10 times. What is the probability that the faces $1,2,3,4,5,6$ will appear respectively $2,3,1,1,1,2$ times (event $A$)?
Solution. The number of outcomes favorable to event $A$ is calculated using formula (1.3.7):

$$
m=\frac{(2+3+1+1+1+2)!}{2!\cdot 3!\cdot 1!\cdot 1!\cdot 1!\cdot 2!}=\frac{10!}{4 \cdot 6}=2 \cdot 3 \cdot 5 \cdot 7 \cdot 8 \cdot 9 \cdot 10
$$

The number of all elementary outcomes in this experiment is $n=6^{10}$, therefore

$$
P(A)=\frac{2 \cdot 3 \cdot 5 \cdot 7 \cdot 8 \cdot 9 \cdot 10}{6^{10}}=\frac{700}{6^{7}} \approx 0.002
$$

## Problems

1. On 5 identical cards, the letters $Б, E, P, C, T$ are written. These cards are randomly arranged in a row. What is the probability that the word БРЕСТ will be formed?
2. In a box, there are 4 blue and 5 red balls. Two balls are randomly drawn from the box. Find the probability that the balls are of different colors.
3. In a team, there are 4 women and 3 men. Four theater tickets are distributed among the team members. What is the probability that 2 women and 2 men will receive the tickets?
4. In a box, there are 10 balls, of which 2 are white, 3 are red, and 5 are blue. Three balls are randomly drawn. Find the probability that all 3 balls are of different colors.
5. On five identical cards, the letters $л, \mathcal{M}, o, o, m$ are written. What is the probability that, when the cards are drawn one by one, the word молот will be formed?
6. From a batch containing 10 items, 3 of which are defective, 3 items are randomly drawn. Find the probability that one of the items in the sample is defective.
7. Out of ten tickets, two are winning tickets. What is the probability that one of the five tickets drawn at random is a winning ticket?

## Answers

1. $1 / 120.2 .5 / 9.3 .18 / 35.4 .0.25.5 .1 / 60.6 .21 / 40.7 .5 / 9$.

## Questions

1. What are permutations called?
2. What formula is used to calculate the number of permutations of $n$ different elements?
3. What are arrangements called?
4. What formula is used to calculate the number of arrangements of $n$ different elements taken $m$ at a time?
5. What are combinations called?
6. What formula is used to calculate the number of combinations of $n$ elements taken $m$ at a time?
7. What equality connects the numbers of permutations, arrangements, and combinations?
8. What formula is used to calculate the number of permutations of $n$ elements if some elements are repeated?
9. What formula defines the number of arrangements of $m$ elements with repetitions from $n$ elements?
10. What formula defines the number of combinations with repetitions of $n$ elements taken $m$ at a time?

## § 1.4. Frequency of an Event. Statistical Definition of Probability

The classical definition of probability assumes that all elementary outcomes are equally likely. The equal likelihood of outcomes is inferred from considerations of symmetry (as in the case of a coin or a die). Problems where symmetry considerations can be applied are rare in practice. In many cases, it is difficult to provide a basis for assuming that all elementary outcomes are equally likely. This led to the need to introduce another definition of probability, called the statistical definition. To give this definition, the concept of relative frequency of an event is introduced first.

The relative frequency of an event, or frequency, is the ratio of the number of trials in which the event occurred to the total number of trials. The frequency of event $A$ is denoted by $W(A)$, then by definition

$$
W(A)=\frac{m}{n}
$$

where $m$ is the number of trials in which event $A$ occurred; $n$ is the total number of trials.

The frequency of an event has the following properties.

1. The frequency of a random event is a number between zero and one:

$$
0<W(A)<1
$$

2. The frequency of a certain event $U$ is one:

$$
W(U)=1
$$

3. The frequency of an impossible event $V$ is zero:

$$
W(V)=0
$$

4. The frequency of the sum of two mutually exclusive events $A$ and $B$ is equal to the sum of the frequencies of these events:

$$
W(A+B)=W(A)+W(B)
$$

Observations have shown that the relative frequency has the property of statistical stability: in different series of multiple trials (in each of which the event may or may not occur), it takes values close to some constant. This constant, which is an objective numerical characteristic of the phenomenon, is considered the probability of the given event.

The probability of an event is the number around which the values of the frequency of the given event group in different series of a large number of trials.

This definition of probability is called the statistical definition.

In the case of the statistical definition of probability, it has the following properties: 1) the probability of a certain event is one; 2) the probability of an impossible event is zero; 3) the probability of a random event is between zero and one; 4) the probability of the sum of two mutually exclusive events is equal to the sum of the probabilities of these events.