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Example 2. Consider the line integral
$$
\int_{L} \frac{-y d x}{x^{2}+y^{2}}+\frac{x d y}{x^{2}+y^{2}}
$$ | Solution. The given expression makes sense everywhere except at the point $O(0,0)$; therefore, we exclude this point. In the remaining part of the plane (which will already be a disconnected region), the coefficients of $d x$ and $d y$ are continuous and have continuous partial derivatives, and they are identically equ... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,629 |
Example 3. Calculate the circulation of the vector $\mathbf{a}=\sqrt{1+x^{2}+y^{2}} \mathbf{i}+y[x y+$ $\left.\ln \left(x+\sqrt{1+x^{2}+y^{2}}\right)\right]$ along the circle $x^{2}+y^{2}=R^{2}$. | Solution. The circulation of the given vector is equal to
$$
\mu=\oint_{L}(\mathbf{2} . d \mathbf{r})=\oint_{L} \sqrt{1+x^{2}+y^{2}} d x+y\left[x y+\ln \left(x+\sqrt{1+x^{2}+y^{2}}\right)\right] d y .
$$
Here
$$
P=\sqrt{1+x^{2}+y^{2}}, \quad Q=x y^{2}+y \ln \left(x+\sqrt{1+x^{2}+y^{2}}\right)
$$
We find the partial... | \frac{\piR^{4}}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,630 |
Example 1 (Electric Field of a Point Charge). Show that the electric field strength $\mathbf{E}$, created by a point charge $q$ placed at the origin:
$$
\mathbf{E}=\frac{q}{r^{3}} \mathbf{r}, \quad r=\sqrt{x^{2}+y^{2}+z^{2}}
$$
is a potential field. | Solution. It will be as follows: show that there exists a function $\varphi(x, y, z)$ such that the relations (2) are satisfied.
In our case, we have
$$
P(x, y, z)=\frac{q x}{r^{3}}, \quad Q(x, y, z)=\frac{q y}{r^{3}}, \quad R(x, y, x)=\frac{q z}{r^{3}}
$$
Since
$$
\frac{\partial}{\partial x}\left(\frac{1}{r}\right... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,631 |
## Example 3. Find the potential of the vector field $\mathbf{a}=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k}$ | Solution. It is easy to see that $\operatorname{rot} \mathbf{a} \equiv 0$, i.e., the given vector field is potential. This field is defined throughout the entire three-dimensional space, which is star-shaped with the center at the origin of coordinates $O(0,0,0)$, so to find its potential, we will use formula (10). Sin... | \varphi(M)=xyz+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,633 |
Example 1. Show that $\operatorname{div}(u \mathbf{a})=u \operatorname{div} \mathbf{a}+(\mathbf{a}, \operatorname{grad} u)$. Here $u-$ is a scalar function, $\mathbf{a}$ is a vector function. | Solution.
In symbolic notation
$$
\operatorname{div}(u a)=(\nabla, u a) .
$$
Considering first the differential character of $\nabla$, we must write
$$
(\nabla, u a)=\left(\nabla, u_{c} \mathbf{a}\right)+\left(\nabla, u_{a_{c}}\right) .
$$
Considering the expression $(\nabla, u_{c} \mathbf{a})$, we can take the co... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,634 |
Example 8. Find the surface integral
$$
I=\int_{\Sigma}\left(\varphi \frac{\partial \psi}{\partial n}-\psi \frac{\partial \varphi}{\partial n}\right) d \sigma
$$
taken over the surface $\Sigma: x^{2}+y^{2}=R^{2}, z=0, z=H(H>0)$, if $\varphi=x^{2}+y^{2}+x+z, \psi=x^{2}+y^{2}+2 z+x$. | Solution. The required integral by the second Green's formula is
$$
I=\iiint_{V}(\varphi \Delta \psi-\psi \Delta \varphi) d v
$$
For the given functions $\varphi$ and $\psi$, we have $\Delta \varphi=4, \Delta \psi=4$, and thus,
$$
I=-4 \iiint_{V} z d u
$$
Transitioning to cylindrical coordinates $x=\rho \cos \varph... | -2\piR^{2}H^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,637 |
Example 1. Find the vector potential $\mathbf{b}=\mathbf{b}(x, y, z)$ for a solenoidal field given by the vector $\mathbf{a} = 2 y \mathbf{i} - z \mathbf{j} + 2 x \mathbf{k}$. | Solution. Generate it in the form
$$
\mathbf{b}=\mathbf{h}(x, y, z)=Q_{1}(x, y, z) \mathbf{j}+R_{1}(x, y, z) \mathbf{k}
$$
where $Q_{1}(x, y, z)$ and $R_{1}(x, y, z)$ are found using formulas (6) and (7). Since in this
| Find the vector potentials of solenoidal fields: | |
| :---: | :---: |
| 256. $a=i+\mathbf{j}+\... | {b}(M)=\frac{1}{3}(2xy+z^{2}){i}+\frac{2}{3}(x^{2}-yz){j}+\frac{1}{3}(2y^{2}+xz){k} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,638 |
Example 1. A vector field is given in cylindrical coordinates $\mathbf{a}(M)=\mathbf{e}_{p}+\varphi \mathbf{e}_{\varphi}$. Find the vector lines of this field. We have | Solution. According to the problem's condition $a_{1}=1, a_{2}=\varphi_{1} a_{3}=0$. By formula (1)
$$
\frac{d \rho}{1}=\frac{\rho d \varphi}{\varphi}=\frac{d z}{0}
$$
Hence
$$
\left\{\begin{array}{l}
z=C_{1} \\
\rho=C_{2 \varphi}
\end{array}\right.
$$
this is a spiral of Archimedes, lying in planes parallel to the... | C_{1},\rho=C_{2\varphi} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,640 |
Example 3. Find the gradient of the scalar field given in spherical coordinates $(r, \theta, \varphi): u=r+\frac{\sin \theta}{r}-\sin \theta \cos \varphi$ | Solution. Using formula (3), we will have
$$
\operatorname{grad} u=\left(1-\frac{\sin \theta}{r^{2}}\right) e_{r}+\frac{\cos \theta}{r}\left(\frac{1}{r}-\cos \varphi\right) e_{\theta}+\frac{\sin \varphi}{r} e_{\varphi}
$$
$3^{\circ}$. Rotor in orthogonal coordinates. Let
$$
a=a_{1}\left(q_{1}, q_{2}, q_{3}\right) \m... | \operatorname{grad}u=(1-\frac{\sin\theta}{r^{2}})e_{r}+\frac{\cos\theta}{r}(\frac{1}{r}-\cos\varphi)e_{\theta}+\frac{\sin\varphi}{r}e_{\varphi} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,642 |
Example 5. Show that the vector field $a=\frac{2 \cos \theta}{r^{3}} \mathbf{e}_{r}+\frac{\sin \theta}{r^{3}} \mathbf{e}_{\theta}$
is solenoidal. | Solution. Using formula (5), we will have
$$
\begin{aligned}
& \operatorname{div}=\frac{1}{r^{2}} \frac{\theta}{\partial r}\left(r^{2} \frac{2 \cos \theta}{r^{3}}\right)+\frac{1}{r \sin \theta} \frac{\theta}{\partial \theta}\left(\sin \theta \frac{\sin \theta}{r^{3}}\right)+0= \\
&=\frac{1}{r^{2}}\left(-\frac{2 \cos \... | 0 | Calculus | proof | Yes | Yes | olympiads | false | 31,643 |
Example 8. Find the potential of the vector field given in cylindrical coordinates: $\mathbf{a}=\left(\frac{\operatorname{arctg} z}{\rho}+\cos \varphi\right) \mathbf{e}_{\rho}-\sin \varphi \mathbf{e}_{\varphi}+\frac{\ln \rho}{1+z^{2}} \mathbf{e}_{z}$. | Solution. Using formula (4), we find
$$
\operatorname{rot} a=\left|\begin{array}{ccc}
\frac{1}{\rho} \mathrm{e}_{\rho} & \mathrm{e}_{\varphi} & \frac{1}{\rho} \mathbf{c}_{z} \\
\frac{\partial}{\partial \rho} & \frac{\partial}{\partial \varphi} & \frac{\partial}{\partial z} \\
\frac{\operatorname{arctg} z}{\rho}+\cos \... | u(\rho,\varphi,z)=\ln\rho\cdot\operatorname{arctg}z+\rho\cos\varphi+C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,645 |
Example 10. Calculate the line integral of the vector field given in cylindrical coordinates: $2=4 \rho \sin \varphi \mathbf{e}_{\rho}+z e^{p} \mathrm{e}_{,}+(\rho+\varphi) \mathbf{e}_{z}$, along the line $L:\left\{\varphi=\frac{\pi}{4}, z=0\right\}$ from point $O\left(0, \frac{\pi}{4}, 0\right)$ to point $A\left(1, \f... | Solution. In this example,
$$
a_{\rho}=4 \rho \sin \varphi, \quad a_{p}=x e^{p}, \quad a_{z}=\rho+\varphi .
$$
By formula (13), the required line integral is
$$
\int_{L}(a, d r)=\int_{L} 4 \pi \sin \varphi d \rho+\rho z e^{\rho} d \varphi+(\rho+\varphi) d z
$$
On the line $\boldsymbol{L}$, we have:
$$
\varphi=\fra... | \sqrt{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,646 |
Example 11. Calculate the line integral of the vector field given in spherical coordinates: $2=e^{r}$ sin $\theta \mathbf{e}_{r}+3 \theta^{2}$ sin $\varphi \mathbf{e}_{\theta}+\tau \varphi \theta \mathbf{e}_{\varphi}$ along the line $L:\left\{r=1, \varphi=\frac{\pi}{2}, 0 \leqslant 0 \leqslant \frac{\pi}{2}\right\}$ in... | Solution. The line $L$ is a quarter circle with the center at the origin and radius $R=1$, located in the yOz plane. The coordinates of the given vector are
$$
a_{r}=e^{r} \sin \theta_{r} \quad a_{s}=3 \theta^{2} \sin \varphi_{,} \quad a_{p}=r \varphi \theta .
$$
According to formula (14), the line integral has the f... | \frac{\pi^{3}}{8} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,647 |
Example 12. Calculate the circulation of the vector field given in cylindrical coordinates: $2=\rho \sin \varphi \mathrm{e}_{\rho}+\rho z \mathrm{e}_{\varphi}+\rho^{3} \mathrm{e}_{z}$, along the curve L: $\{\rho=\sin \varphi, z=0,0 \leqslant \varphi \leqslant \pi\}$ directly and using Stokes' theorem. | Solution. Coordinates of the given vector
$$
a_{p}=\rho \sin \varphi, \quad a_{v}=\rho z, \quad a_{n}=\rho^{3}
$$
The contour $L$ represents a closed curve located in the plane $z=0$ (Fig. 41).
1) Direct calculation of circulation.
Substituting the coordinates of the vector into formula (13), we get
$$
\boldsymbol... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,648 |
Example 13. Calculate the circulation of the vector field given in spherical coordinates: $2=r \mathbf{e}_{r}+(R+r) \sin \theta \mathbf{e}_{\varphi}$, along the circle $L:\{r=$ $\left.R, \theta=\frac{\pi}{2}\right\}$ in the direction of increasing angle $\varphi$ directly and using Stokes' theorem. | Solution. In the given example
$$
a_{r}=r_{1} \quad a_{9}=0, a_{\varphi}=(R+r) \sin \theta
$$
1) Direct calculation of circulation. According to formula (14), the desired circulation is
$$
\square=\oint_{L} r d r+(R+r) \sin \theta \cdot r \sin \theta d \varphi=\oint_{L} r d r+r(R+r) \sin ^{2} \theta d \varphi
$$
On... | 4\piR^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,649 |
1. In a computer program written in Turbo Pascal, the function $\operatorname{Random}(x)$ is used, generating integer random numbers from 1 to $x$. What is the probability that when this function is executed, a number divisible by 5 will appear if $x=100?$ | Solution. Let's denote the event: $A$ - a number divisible by 5 will appear when $x=100$. We will find the probability of event $A$ by applying formula (1).
When $x=100$, any of the 100 existing integers can appear, so the total number of outcomes of the trial is $n=100$.
To find the number of outcomes of the trial f... | 0.2 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 31,650 |
3. A child is playing with letters from a cut-out alphabet. What is the probability that, by arranging the letters К, И, Р, Д, А, Н, З, П in a row, they will form the word ПРАЗДНИК? | S o l u t i o n. Let's denote the event: $A$ - the child will form the word IIPAZDNIC.
Let's find the probability of event $A$, applying formula (1).
The total number of $n$ outcomes of the trial will be obtained using the formulas of combination theory. There are 8 elements - 8 letters; in the formation of different... | \frac{1}{40320} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,651 |
4. There are 8 cards; one side of each is clean, while the other side has the letters: И, Я, Л, З, Г, О, О, О printed on them. The cards are placed on the table with the clean side up, shuffled, and then sequentially flipped over one by one. What is the probability that the letters will form the word ЗОоЛОГИЯ when they... | Solution. Let us denote the event: $\boldsymbol{B}$ - the word $300-$ logy will be formed.
We will find the probability of event $B$ by applying formula (1). The numbers $m$ and $n$, included in this formula, will be determined using the formulas of combination theory.
The total number of outcomes of the experiment, ... | \frac{1}{6720} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,652 |
5. A music lover, having numbered five newly listened to compact discs with the digits $1,2,3,4,5$, placed them in the cassette player in a random order. What is the probability that discs №1 and №2 will be placed next to each other in the cassette player and in ascending order of their numbers? | Solution. Let's denote the event: $A$ - compact discs №1 and №2 will be placed next to each other in the cassette holder and in ascending order of their numbers.
We will find the probability of event $A$ by applying formula (1).
The total number of $n$ outcomes of the experiment can be obtained using the formulas of ... | \frac{1}{5} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,653 |
6. From a deck of cards, 4 aces and 4 kings were removed. These cards were shuffled and laid out in a row. What is the probability that all 4 kings will be located next to each other? | S o l u t i o n. Let's denote the event: $A$ - 4 kings will be located next to each other. The probability of event $A$ will be found using formula (1).
The total number of $n$ possible outcomes of the experiment will be obtained similarly to how it is done in problem 5. There are 8 elements in total - 8 cards. These ... | \frac{1}{14} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,654 |
7. Seedlings of two varieties of black currant, 6 of the Selyanchenskaya variety and 8 of the Vologda variety, are prepared for planting in a garden plot and are accidentally mixed. What is the probability that the first 3 currant seedlings planted will be of the Selyanchenskaya variety? | S o l u t i o n. Let's denote the event: $A$ - the first 3 currant seedlings planted are of the Selchenkaya variety.
We will find the probability of event $A$ by applying formula (1). The numbers $m$ and $n$, which are part of this formula, will be obtained using the formulas of combination theory.
. The numbers $m$ and $n$, which are part of this formula, will be obtained using the formulas of combination theory.
There are a total of $N$ el... | P(A)=\frac{C_{k}^{r}\cdotC_{N-k}^{-r}}{C_{N}^{}} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,657 |
11. A group of 11 people, including Ivanov and Petrov, are seated around a round table in a random order. Find the probability that there will be 3 people sitting between Ivanov and Petrov. | Solution. Let's denote the event: $A$ - there will be 3 people sitting between Ivanov and Petrov at the table.
We will find the probability of event $A$ by applying formula (1). The numbers $m$ and $n$, which are part of this formula, will be obtained using the formulas of combination theory. There are 11 elements - 1... | \frac{1}{10} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,658 |
14. The knut ciphers in the library catalog consist of six digits and do not start with the digit 0. A reader is looking for the cipher of the book they need in the catalog. What is the probability that all the digits of the cipher will be different? | S o l u t i o n. Let us denote the event: $\boldsymbol{A}$ - all the digits of the code the reader is looking for are different.
We will find the probability of event $A$ by applying formula (1).
Let us determine the numbers $m$ and $n$ that are part of this formula.
There are ten elements in total - ten digits: $0,... | 0.13608 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,659 |
15. On the table, there are two stacks of notebooks. In the first stack, there are 5 notebooks with blue covers, and in the second stack, there are 5 notebooks with red covers. The notebooks in each of these stacks are numbered with the digits $1,2,3,4,5$ and are arranged in a random order of numbers. A student takes o... | Solution. The experiment consists of extracting two notebooks from stacks. We will construct a fifth-order square matrix characterizing all 25 equally possible outcomes of this experiment, representing a complete group of mutually exclusive events:
$$
D=\left(\begin{array}{lllll}
11 & 12 & 13 & 14 & 15 \\
21 & 22 & 23... | P(A)=\frac{1}{5},P(B)=\frac{1}{9},P(C)=\frac{1}{5} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,660 |
16. When determining the germination of a batch of seeds, a sample of 1000 units was taken. Out of the selected seeds, 90 did not germinate. What is the relative frequency of the appearance of viable seeds | S o l u t i o n. Let's denote the event: $A$ - a viable seed has been selected. We will find the relative frequency of event $A$ by applying formula (5). The total number of trials conducted is $n=1000$. The number of trials in which event $A$ occurred is $m=1000-90=910$.
The relative frequency of event $A$ is $W(A)=\... | 0.91 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,661 |
17. For conducting research on a certain field, a random sample of 200 wheat ears was taken. The relative frequency (frequency) of ears having 12 spikelets per ear turned out to be 0.125, and for 18 spikelets it was 0.05. Find the frequencies of ears having 12 and 18 spikelets in this sample. | S o l u t i o n. Let's consider the events: $A$ - a spike with 12 spikelets is taken; $B$ - a spike with 18 spikelets is taken.
Let's find the frequencies $m_{1}$ and $m_{2}$ of events $A$ and $B$, using formula (5).
Denote by $W_{1}(A)=\frac{m_{1}}{n}$ the relative frequency of event $A$, and by $W_{2}(B)=\frac{m_{2... | 25,10 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,662 |
19. A smaller segment $\ell$ with a length of 15 cm is placed on a larger segment $L$ with a length of 40 cm. Find the probability that a point randomly placed on the larger segment will also fall on the smaller segment. It is assumed that the probability of the point falling on segment $\ell$ is proportional to the le... | Solution. Let event $A$ be the point randomly placed on segment $L$ also falls on segment $\ell$.
We will find the probability of event $A$ by applying formula (6):
$$
P(A)=\frac{15}{40}=\frac{3}{8} .
$$ | \frac{3}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,663 |
21. The Meeting Problem. Two comrades agreed to meet at a certain place between 12 o'clock and half past one in the afternoon. The one who arrives first waits for the other for 20 minutes, after which he leaves. Find the probability that the comrades will meet, if each of them randomly chooses the moment of their arriv... | Solution. Let event $A$ be the meeting of friends.
We will find the probability of event $A$ by applying formula (7).
Let the arrival time of one of them be denoted by $x$, in minutes, and the arrival time of the other by $y$, in minutes. For the meeting to occur, it is necessary and sufficient that the condition $|x... | \frac{8}{9} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,664 |
22. The coefficients $p$ and $q$ of the quadratic equation $x^{2} + p x + q = 0$ are chosen randomly in the interval $(0 ; 2)$. What is the probability that the roots of this equation will be real numbers? | Solution. Let event $A$ be that the roots of the given equation are real numbers.
We will find the probability of event $A$ by applying formula (7). Let the coefficients $p$ and $q$ of the quadratic equation be randomly chosen numbers. Their possible values are: $0 \leq p \leq 2 ; 0 \leq q \leq 2$. Represent $p$ and $... | \frac{1}{6} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,665 |
24. In a money and goods lottery, for every 1000 tickets, there are 5 monetary and 20 goods prizes. What is the probability of winning on one ticket? | S o l u t i o n. Consider the events:
$A_{1}$ - a material prize for one ticket;
$A_{2}$ - a monetary prize for one ticket;
$A$ - any prize for one ticket.
$A_{1}$ and $A_{2}$ are mutually exclusive events. Event $A$ consists in the occurrence of either event $A_{1}$ or event $A_{2}$ (it does not matter which); thi... | 0.025 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,666 |
25. For an exam answer, a student can receive one of the following grades: $5,4,3,2$. The probability that a student will receive a grade of 5 is 0.3; a grade of $4-0.4$; a grade of $3-0.2$ and a grade of $2-0.1$. Which of the named events form a complete group of mutually exclusive events? What event is opposite to th... | S olution. Consider the events: $A_{1}, A_{2}, A_{3}, A_{4}$ - the student will receive, respectively, grades: $5,4,3,2$;
$A$ - the student will receive one of these grades: either 5, or 4, or 3, or 2.
The probabilities of events $A_{1}, A_{2}, A_{3}, A_{4}$ are: $P\left(A_{1}\right)=0.3 ; P\left(A_{2}\right)=$ $=0.4... | 0.7 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,667 |
30. In the green cutting department of the fruit experimental station, 20 green cuttings were prepared for planting in the greenhouse, including 8 cuttings of winter-hardy apricot variety 9-114, and the rest are cuttings of plum variety Eurasia 21. Three cuttings were randomly selected. Find the probability that at lea... | S o l u t i o n. This problem is of the same type as problem 29. For its rational solution, we recommend applying the method considered in the second part of problem 29.
Consider the events:
$A$ - at least one plum cutting is selected;
$\bar{A}$ - no plum cutting is selected.
The probability of event $\bar{A}$ is
... | \frac{46}{57}\approx0.8070 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,668 |
31. In the case, there are 20 cassettes with disco music recordings and 10 - with techno music recordings. The DJ randomly takes two cassettes one after another. What is the probability that
1) the first cassette has disco music recorded on it;
2) the second cassette also has disco music recorded on it. Consider two ca... | Solution. Consider the events:
$A$ - disco music is recorded on the first cassette;
$B$ - disco music is recorded on the second cassette.
1) Let's find the probability of event $A$, using the classical definition of probability. The total number of outcomes of the experiment $n=30$; the number of outcomes of the exp... | \frac{2}{3},\frac{2}{3},\frac{19}{29} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,669 |
32. Using the original data from the previous problem, find the probability that music in the disco style is recorded on two randomly taken cassettes in succession. Consider two cases: the DJ, before taking the second cassette,
a) returns the first cassette to the case;
b) does not return the first cassette to the ca... | S o l u t i o n. Let's consider the events:
$A$ - disco music is recorded on the first cassette;
$\boldsymbol{B}$ - disco music is recorded on the second cassette;
C - disco music is recorded on two consecutively taken cassettes.
Event $C$ consists of disco music being recorded on both the first and second cassette... | \frac{4}{9}\approx0.4444\frac{20}{30}\cdot\frac{19}{29}\approx0.4368 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,670 |
33. Randomly mixed are seedlings of two tomato varieties: 9 seedlings of the variety White Naliv and 7 - of the variety Verlioka. Find the probability that the first three tomato plants, planted one after another, are seedlings of the variety White Naliv. | S o l u t i o n. The test consists of planting one tomato seedling. Let's consider the events:
$A_{1}$ - the first planted bush is a tomato seedling of the White Naliv variety;
$A_{2}$ - the second planted bush is a tomato seedling of the White Naliv variety;
$A_{3}$ - the third planted bush is a tomato seedling of ... | 0.15 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,671 |
35. In an urn, there are 4 white, 6 black, and 5 red balls. Two balls are drawn at random one after the other. Find the probability that both balls are of the same color. | Solution. Consider the events:
$A_{1}$ - the first ball drawn is white;
$B_{1}$ - the second ball drawn is white;
$A_{2}$ - the first ball drawn is black;
$B_{2}$ - the second ball drawn is black;
$A_{3}$ - the first ball drawn is red;
$B_{3}$ - the second ball drawn is red;
$C$ - two balls of the same color are... | \frac{31}{105} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,673 |
37. A regular triangle is inscribed in a circle of radius $R$. What is the probability that four randomly placed points within this circle will be inside the triangle? | Solution. Consider the events:
$A_{i}$ - a randomly placed point in the circle will be inside the equilateral triangle inscribed in this circle ( $i=1,2,3,4$ ).
B - four randomly placed points in the circle will be inside the equilateral triangle inscribed in this circle.
Event $B$ consists of the first, second, thi... | 0.029 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,674 |
39. A batch of one hundred parts is subject to selective control. The condition for the rejection of the entire batch is the presence of at least one defective part among the four inspected. What is the probability that the batch will not be accepted if it contains $3 \%$ defective parts? | S o l u t i o n. a) Let's denote the events:
$A_{i}$ - the inspected part is defective ( $i=1,2,3,4$ );
$\bar{A}_{i}$ - the inspected part is not defective;
$A$ - the batch of parts will not be accepted (at least one defective part will be found among the four inspected);
$\bar{A}$ - the batch of parts will be acce... | 0.1164 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,675 |
41. Let $A_{1}, A_{2}, \ldots, A_{i}, \ldots, A_{n}$ be $n$ independent events in total. The probabilities of each event $A_{i}$ occurring are $p_{i}$ $(i=1,2, \ldots, n)$. Suppose that as a result of the trial, all events or some of them may occur, or none of them may occur. Find the probability of at least one of the... | S o l u t i o n. In addition to the $\boldsymbol{n}$ independent events $A_{i}$, consider their opposites $\bar{A}_{i}$, as follows:
$A$ - the occurrence of at least one of the events $A_{i}$;
$\bar{A}$ - the non-occurrence of any of the events $A_{i}$.
Since $P\left(A_{i}\right)=p_{i}$, then $P\left(\bar{A}_{i}\rig... | P(A)=1-q^{n} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,677 |
42. The probability of the establishment of a stable snow cover in a certain area from October is 0.2. What is the probability that in the next two years in this area a stable snow cover from October
a) will not establish once;
b) will establish at least once? | Solution. Let's denote the events:
$A_{1}$ - a stable snow cover from October will establish in the first year;
$A_{2}$ - a stable snow cover from October will establish in the second year;
$\bar{A}_{1}$ - a stable snow cover from October will not establish in the first year;
$\bar{A}_{2}$ - a stable snow cover fro... | 0.64 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,678 |
43. The subscriber has forgotten the last digit of the phone number and therefore dials it at random. What is the probability that they will have to dial the number no more than three times? | S o l u t i o n. Let us denote the event:
C - the subscriber will have to dial the number no more than three times.
This event consists in the subscriber having to dial the number either once, or twice, or three times. Let us consider the following events:
$C_{1}$ - the subscriber will dial the number once;
$C_{2}$... | 0.3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,679 |
45. How many numbers need to be selected from a table of random numbers to be confident with a probability of at least 0.9 that among them there is at least one even number? | Solution. Let $n$ be the required number of random numbers. Consider the events:
$A_{k}$ - one randomly selected number is even $(k=1,2, \ldots, n)$;
$\overline{A_{k}}$ - one randomly selected number is odd;
$B$ - among $n$ random numbers there will be at least one even;
$\bar{B}$ - among $n$ random numbers there w... | 4 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,680 |
46. Electric lamps are manufactured at three plants. The first plant produces $35 \%$ of the total number of lamps, the second - $50 \%$, and the third $15 \%$. The production of the first plant contains $70 \%$ standard lamps, the second $80 \%$, and the third $90 \%$. The products of all three plants are supplied to ... | S o l u t i o n. Let's consider the events:
$B_{1}$ - a randomly selected lamp is manufactured by the first factory;
$B_{2}$ - a randomly selected lamp is manufactured by the second factory;
$B_{3}$ - a randomly selected lamp is manufactured by the third factory;
$C_{1}$ - a randomly selected lamp is manufactured b... | 0.78 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 31,681 |
48. The number of trucks passing by a gas station on a highway is to the number of passenger cars passing by the same highway as $3: 2$. It is known that on average, 1 out of 30 trucks and 2 out of 45 passenger cars pull up to the gas station for refueling. What is the probability that a vehicle arriving at the gas sta... | Solution. Consider the events:
$B_{1}$ - a truck has pulled up to the gas station;
$B_{2}$ - a car has pulled up to the gas station;
$A$ - the vehicle that has pulled up to the gas station will be refueled.
Given that the number of trucks is to the number of cars as $3: 2$, we find the probabilities of the hypothese... | 0.0378 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 31,682 |
49. On the user's computer desktop, there are two folders with files. In the first folder, there are 16 files, and 4 of them are less than 500 kilobytes in size. In the second folder, there are 20 files, and 5 of them are less than 500 kilobytes in size. Without considering the file sizes, the user moves one file from ... | S o l u t i o n. Let's consider the following assumptions about the user's options for transferring files from the first folder to the second (hypotheses):
$B_{1}$ - a file less than $500 \mathrm{Kb}$ in size is transferred from the first folder to the second;
$B_{2}$ - a file of at least $500 \mathrm{Kb}$ in size is... | \frac{1}{4} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,683 |
52. There are 10 identical urns, 9 of which contain 2 black and 2 white balls, and one contains 5 white and 1 black ball. A ball is drawn from a randomly selected urn. The drawn ball turned out to be white. What is the probability that this ball was drawn from the urn containing 5 white balls? | S o l u t i o n. There are 2 groups of urns with different compositions of balls; 9 of the existing urns belong to the first group, one urn - to the second group. The experiment consists of drawing a ball from a randomly selected urn. Let's consider the hypotheses:
$B_{1}$ - an urn from the first group is selected;
$... | 0.15625 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,684 |
53. The first machine supplies 80% of the parts for assembly, while the second machine supplies 20% of the same parts. The defect rate on the first machine is 1%, and on the second machine, it is 5%. A checked part turned out to be defective. Is it more likely that this part was manufactured on the first machine or on ... | Solution. The test consists of checking the quality of a part. Consider the events:
$B_{1}$ - the part checked was manufactured on the first machine;
$B_{2}$ - the part checked was manufactured on the second machine;
$A$ - the part checked is defective.
The unconditional probabilities of the hypotheses - events $B_... | P_{A}(B_{2})>P_{A}(B_{1}) | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 31,685 |
55. Calves are delivered to a fattening complex from three farms. The number of calves from the first farm is twice as many as from the second, and the number from the second farm is three times as many as from the third. The first farm supplies $15 \%$ of the calves weighing more than 300 kg. The second and third farm... | S o l u t i o n. The test consists of weighing a randomly selected calf from among those received at the fattening complex.
Consider the events:
$B_{1}, B_{2}, B_{3}$ - a calf from the 1st, 2nd, and 3rd farm, respectively, has been selected;
$A$ - a randomly selected calf has a live weight exceeding $300 \mathrm{kr}... | 0.175 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,687 |
56. A user has three computer floppy disks made by companies K, L, and M, one disk from each of these companies, and the company stamps on the disks are missing. Two of the three disks turned out to be defective. What is the probability that the defective disks are from companies L and M, if the defect rate in the prod... | The problem is solved. Let's denote the event: $A$ - two floppy disks are defective.
Event $A$ can only occur under the condition of the appearance of one of the incompatible events $B_{i}$ (hypotheses), which form a complete group. Let's consider the hypotheses:
$B_{1}$ - the floppy disks from firms $\mathrm{K}$ and... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,688 | |
63. A large batch of car tires contains $1.5\%$ defects. What should be the volume of a random sample so that the probability of finding at least one defective car tire in it would be more than $0.92?$ | S o l u t i o n. Let the volume of the random sample be denoted by $n$. The experiment consists of checking car tires. Event $A$, which may or may not occur in each trial, is that the checked car tire turns out to be defective. The probability $p=0.015$; the probability $q=0.985$. Consider the events:
C - the sample o... | 168 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,689 |
64. The probability of failure of each device during testing is 0.2. How many such devices need to be tested to assert with a probability of at least 0.9 that at least two devices will fail? | S o l u t i o n. Let the required number of devices be denoted by $n$. The event $A$, which may or may not occur in each trial, consists of a device failing. The probability $p=0.2$; the probability $q=0.8$.
Consider the events:
$B$ - at least two devices fail;
$\bar{B}$ - fewer than two devices fail.
The event $\b... | 18 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,690 |
68. For the red clover variety Perm local, on average $84\%$ of the plants are late maturing. What is the probability that 52 plants out of 60 clover plants, randomly selected, will be late maturing? | S o l u t i o n. In each trial, it is determined whether the clover plant is late-maturing. The total number of independent trials is $n=60$. Event $A$, which may or may not occur in each trial, consists in the fact that the selected plant is late-maturing. The probability of event $A$ occurring in each trial is $p=0.8... | 0.1201 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,691 |
70. An experimental plot is sown with seeds of smooth brome. On one of the plots of this plot, the grass stand contains $0.4\%$ of weeds, namely white cockle and broad-leaved weeds. What is the probability that among 125 plants on this plot, randomly selected, there are
a) exactly 3 weeds;
b) no more than three weeds... | S o l u t i o n. The test consists of checking one plant. The total number of independent trials is $n=125$. In each trial, event $A$, which consists of the selected plant being a weed, may or may not occur. The probability $p=0.004$ of event $A$ occurring in each trial is very small, and $\lambda=n p=125 \cdot 0.004=0... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,692 |
74. The probability that the total length of flax plants of variety $A$ is $75-84$ cm is 0.6. What is the probability that among 300 flax plants of this variety, the relative frequency of plants of such length will deviate in absolute value from the probability of the appearance of plants of such length by no more than... | The problem is solved. According to the conditions $n=300 ; \varepsilon=0.05, p=0.6$; we find $q=0.4$.
The required probability will be found using formula (23):
$$
P\left(\left|\frac{m}{n}-0.6\right| \leq 0.05\right) \approx 2 \Phi\left(0.05 \sqrt{\frac{300}{0.6 \cdot 0.4}}\right) \approx 2 \Phi(1.77)=2 \cdot 0.4616... | 0.9232 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,693 |
75. The probability of infection by powdery smut for a certain variety of millet is 0.3. A random sample of 90 millet panicles of this variety is selected. Find the limit of the absolute value of the deviation of the relative frequency of infected panicles from the probability $p=0.3$, if this limit must be guaranteed ... | Solution. According to the condition $p=0.3 ; q=0.7 ; n=90$; $P\left(\left|\frac{m}{n}-0.3\right| \leq \varepsilon\right)=0.9836$. We need to find $\varepsilon$.
According to formula (23)
$$
P\left(\left|\frac{m}{n}-p\right| \leq \varepsilon\right) \approx 2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)=0.9836
$$... | 0.1159 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,694 |
76. When crossing two varieties of peas (with yellow and green seeds), G. Mendel found that in the second generation, the probability of green seeds appearing is 0.25. How many pea seeds need to be taken to expect with a probability of 0.9770 that the relative frequency of green seeds will deviate (in absolute value) f... | Solution. According to the condition $p=0.25 ; \varepsilon=0.02 ;$ we find $q=0.75$. It is known that $P\left(\left|\frac{m}{n}-0.25\right| \leq 0.02\right)=0.997$. We need to find the number of seeds $n$.
In accordance with formula (23) we have
$$
P\left(\left|\frac{m}{n}-p\right| \leq \varepsilon\right) \approx 2 \... | 4107 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,695 |
78. Find the distribution series of the random variable $X$ - the number of times 6 points appear when rolling a die once. | Solution. The random variable $X$ is the number of times a 6 is rolled in a single throw of a fair die. As a result of the experiment, it can take one of the following two possible values: $x_{1}=0 ; x_{2}=1$.
Events $A_{1}, A_{2}, \ldots, A_{6}$ - rolling, respectively, 1, 2, ..., 6 points in a single throw of a die ... | X:\begin{pmatrix}x_{i}&0&1\\p_{i}&5/6&1/6\end{pmatrix} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,696 |
80. The game involves throwing rings onto pegs. The player receives four rings and throws one of these rings at a time until the first successful hit on a peg. The probability of hitting the peg with each throw is 0.1. Find the distribution series of the random variable $X$ - the number of unused rings by the player. | S o l u t i o n. A game consisting of throwing four rings onto a peg represents the implementation of independent trials, each of which may or may not result in event $A$ - the ring landing on the peg. The probability of a hit in one throw is $p=0.1$; the probability of a miss in one throw is $q=1-p=0.9$.
Having four ... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,697 |
81. A farmer has 15 cows, 5 of which produce more than 4500 liters of milk per year. Three cows are randomly selected from those owned by the farmer. Find the distribution law of the random variable $X$ representing the number of cows with the specified high milk yield among the selected ones. | Solution. The random variable $X$ is the number of cows among the selected ones that produce more than 4500 liters of milk per year. This variable can take one of the following possible values as a result of the trial: $x_{1}=0 ; x_{2}=1 ; x_{3}=2 ; x_{4}=3$. The probabilities of the possible values of the variable $X$... | X:\begin{pmatrix}x_{i}&0&1&2&3\\p_{i}&24/91&45/91&20/91&2/91\end{pmatrix} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,698 |
86. A discrete random variable is given by the distribution series:
$$
X: \begin{array}{ccccc}
x_{i} & -2 & 0 & 3 & 7 \\
p_{i} & 0.3 & 0.1 & 0.5 & 0.1^{\circ}
\end{array}
$$
Find the distribution function $F(x)$ and plot the graph of this function. | S o l u t i o n. 1) On the interval $-\infty < x < 7$
\end{array}\right.
$$
The graph of this function is shown in Fig. 13.
Fig. 13 | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,699 |
87. A discrete random variable $X$ is given by the distribution function
$$
F(x)=\left\{\begin{array}{lll}
0 & \text { if } & x \leq 2 \\
0.5 & \text { if } & 2 < x \leq 6 \\
1 & \text { if } & x > 8
\end{array}\right.
$$
Find the probability that $X$ will take a value not less than 4 and less than 8. | Solution. The required probability $P(4 \leq x<8)$ will be found using formula (28):
$$
P(4 \leq x<8)=F(8)-F(4)=0.7-0.5=0.2 .
$$
## 4.2. Numerical Characteristics of Discrete Random Variables
The mathematical expectation of a discrete random variable $X$, having a finite number of possible values, is equal to
$$
M(... | 0.2 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,700 |
88. A discrete random variable $X$ is given by the distribution series:
$$
X: \begin{array}{cccc}
x_{i} & 1 & 2 & 4 \\
p_{i} & 0.1 & 0.3 & 0.6^{\circ}
\end{array}
$$
Find its mathematical expectation, variance, and standard deviation. | Solution. The expected value of the random variable $X$ will be found using formula (29):
$$
M(X)=x_{1} p_{1}+x_{2} p_{2}+x_{3} p_{3}=1 \cdot 0.1+2 \cdot 0.3+4 \cdot 0.6=3.1
$$
The variance of $X$ will be calculated using formula (37) and formula (38). Applying formula (37), we get
$$
\begin{aligned}
& D(X)=\sum_{i=... | 1.1358 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,701 |
89. As a result of processing data from long-term observations, distributions of random variables $X$ and $Y$ have been obtained - the number of households in each of two districts of the region where the yield of spring cereals can exceed 35 centners/ha.
For the first district of the region:
$$
X: \begin{array}{cccc... | Solution. a) Let's find all possible values of the random variable $Z=X+Y$ and the probabilities $P\left(Z=z_{k}\right)$ of these values, using reasoning similar to that presented in problems 79 and 80. As a result, we will obtain the following distribution of the variable $Z$:
=3,\,D(Z)=0.52 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,702 |
90. Find the mathematical expectation of the random variable $Z=2 X+4 Y+5$, if the mathematical expectations of $X$ and $Y$ are known: $M(X)=3, M(Y)=5$. | Solution. Using the properties of mathematical expectation (formulas (31) - (33)), we get:
$$
\begin{aligned}
& M(Z)=M(2 X+4 Y+5)=2 M(X)+4 M(Y)+5= \\
& =2 \cdot 3+4 \cdot 5+5=31
\end{aligned}
$$ | 31 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,703 |
91. A discrete random variable $X$ has three possible values: $x_{1}=1, x_{2}$, and $x_{3}$, with $x_{1}<x_{2}<x_{3}$. The probabilities that $X$ will take the values $x_{1}$ and $x_{2}$ are 0.3 and 0.2, respectively. The expected value of this variable is $M(X)=2.2$, and the variance is $D(X)=0.76$. Find the distribut... | Solution. Since the sum of the probabilities of all possible values of a discrete random variable is equal to one, the unknown probability $p_{3}=P\left(X=x_{3}\right)=1-\left(p_{1}+p_{2}\right)=1-(0.3+0.2)=0.5$.
Then the distribution law of the variable $X$ is given as:
$$
X: \begin{array}{cccc}
x_{i} & 1 & x_{2} & ... | X:\begin{pmatrix}x_{i}&1&2&3\\p_{i}&0.3&0.2&0.5\end{pmatrix} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,704 |
93. Prove that for independent random variables $X$ and $Y$, the following equality holds: $D(X-Y)=D(X)+D(Y)$. | Solution. Using formulas (41) and (42), we get:
$$
\begin{aligned}
& D(X-Y)=D(X+(-Y))=D(X)+D((-1) \cdot Y)= \\
& =D(X)+(-1)^{2} D(Y)=D(X)+D(Y)
\end{aligned}
$$
which is what we needed to prove. | proof | Algebra | proof | Yes | Yes | olympiads | false | 31,705 |
94. Random variables $X$ and $Y$ are independent. The variances of these variables are known: $D(X)=5 ; D(Y)=9$. Find the variance of the random variable $Z=2X-Y+5$. | S o l u t i o n. Using the properties of variance reflected in formulas (40), (41), and (42), we get
$$
D(Z)=D(2 X-Y+5)=2^{2} \cdot D(X)+(-1)^{2} \cdot D(Y)=4 \cdot 5+9=29
$$ | 29 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,706 |
96. The probability of the engine starting with one attempt is constant and equal to p. Formulate the distribution series of the random variable $X$ - the number of engine starts per one attempt. Find the mathematical expectation and variance of $X$. | S o l u t i o n. The test consists of an attempt to start the engine. In the test, event $A$ - the engine starting - may or may not occur. The probability of event $A$ occurring in one test is $p$; the probability of the opposite event $\bar{A}$ - the engine will not start, is $q=1-p$. The random variable $X$ as a resu... | M(X)=p,\D(X)=pq | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,707 |
97. An urn contains 5 white and 10 black balls. A ball is randomly drawn from the urn. Find the distribution of the random variable $X$ - the number of white balls drawn. Calculate the mathematical expectation, variance, and standard deviation of this random variable. | S o l u t i o n. The experiment consists of drawing one ball. In the experiment, event $A$ - the extraction of a white ball - may or may not occur. The probability of event $A$ occurring is found using formula (1): $P(A)=\frac{m}{n}=\frac{5}{15}=\frac{1}{3}$.
Let $P(A)=p, P(\bar{A})=1-p=q$. Thus, $p=1 / 3$, $q=2 / 3$.... | M(X)=\frac{1}{3},D(X)=\frac{2}{9},\sigma(X)=\frac{\sqrt{2}}{3} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,708 |
98. Find the distribution series of the random variable $X$ - the number of times a six is rolled in three throws of a die. Calculate the mathematical expectation, variance, and standard deviation of this random variable. | S o l u t i o n. The experiment consists of one roll of a fair die. The total number of trials is $n=3$. The trials are independent, as the result of each previous trial does not affect the result of the subsequent one. In each trial, event $A$ - the appearance of a six - may or may not occur. The probability of event ... | M(X)=\frac{1}{2},D(X)=\frac{5}{12},\sigma(X)=\frac{\sqrt{15}}{6} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,709 |
99. In the rye variety Tulunskaya Green-grained, during the testing of a certain population, it was found that along with green-grained plants, there are also yellow-grained plants, and 25% of the plants are yellow-grained. Four rye plants from this population were randomly selected. Find the distribution law of the ra... | S o l u t i o n. The test consists in determining the color of the grains of one plant. The total number of independent trials is $n=4$. In each trial, event $A$ may or may not occur, which consists in the fact that a randomly selected plant is green-grained. The probability of event $A$ occurring in each trial is $p=0... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,710 |
101. One of the most important characteristics of human blood is the Rh factor. The "Rh positive" gene is dominant over the "Rh negative" gene.
In one of the surveyed populations, the probability that a person has a positive Rh factor is 0.91. For the random variable $X$ - the number of Rh-negative people among 1000 p... | The random variable $X$ under consideration - the number of Rh-negative individuals among 1000 people in the examined population, is distributed according to the binomial law.
The total number of trials $n=1000$. In each trial, event $A$ - a person is Rh-negative, may or may not occur. The opposite event $\bar{A}$ - a... | x_{i}\in[70;110] | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,712 |
103. Within an hour, a switchboard installed to connect telephone extensions in the offices of a trading company receives on average 90 calls. Assuming that the number of calls on any time interval follows a Poisson distribution,
a) construct the distribution series of the random variable $X$ - the number of calls rec... | Solution. The random variable $X$ - the number of calls received by the switchboard in four minutes, is distributed according to the Poisson law. On average, 90 calls are received by the switchboard per hour, so the average number of calls received by the switchboard in four minutes is $\frac{90}{60} \cdot 4=6$. This m... | 0.9380 | Other | math-word-problem | Yes | Yes | olympiads | false | 31,713 |
104. In the observations of Rutherford and Geiger, a radioactive substance emitted on average $3.87 \alpha$-particles over a period of 7.5 seconds. Find the probability that this substance will not emit any $\alpha$-particles in one second. | Solution. The random variable $X$ - the number of $\alpha$-particles emitted by a substance in one second, is distributed according to the Poisson law. The mathematical expectation of the variable $X$ is $\lambda=3.87 / 7.5=0.516$. The required probability that $X=0$ is $P=e^{-\lambda}=e^{-0.515}=0.5669$. | 0.5669 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,714 |
105. The average density of pathogenic microorganisms in one cubic meter of air is 100. A sample of 2 dm $^{3}$ of air is taken. Find the probability that at least one microbe will be found in the sample. | Solution. Let the random variable $X$ be the number of pathogenic microorganisms found in 2 dm $^{3}$ of air. We adopt the hypothesis of a Poisson distribution of the number of microorganisms that can be detected in this volume. The expected value of $X$ is $\lambda=\frac{100}{1000} \cdot 2=0.2$. The probability that a... | 0.181 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 31,715 |
106. Prove that the Poisson distribution is a limiting case of the binomial distribution (as $n \rightarrow \infty$ and $n p=\lambda=$ const). | Solution. The binomial distribution is defined by the Bernoulli formula (18).
From the equality $n p=\lambda$ we find $\dot{p}=\frac{\lambda}{n}$, hence $q=1-p=1-\frac{\lambda}{n}$.
Substitute these values of $p$ and $q$ into the Bernoulli formula and perform the transformations:
$$
\begin{aligned}
& P_{n}(m)=C_{n}^... | proof | Algebra | proof | Yes | Yes | olympiads | false | 31,716 |
107. The random variable $X$ is given by the distribution function
$$
F(x)=\left\{\begin{array}{lll}
0 & \text { if } & x \leq 2 \\
(x-2)^{2} & \text { if } & 2 < x \leq 3 \\
1 & \text { if } & x > 3
\end{array}\right.
$$
Find:
a) the probability density function $f(x)$;
b) the graphs of the functions $F(x)$ and $f... | Solution. a) The probability density function $f(x)$ is the derivative of the distribution function $F(x)$, so for $x \leq 2$ and $x > 3$, the function $f(x)=0$. Therefore, the probability density function is characterized by the expression:
$$
f(x)=\left\{\begin{array}{lll}
0 & \text { if } & x \leq 2 \\
2(x-2) & \te... | 0.24 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,717 |
109. The random variable $X$ is given by the probability density function:
$$
f(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x \leq \frac{\pi}{4} \\
2 \sin 2 x & \text { if } & \frac{\pi}{4} < x \leq \frac{\pi}{2} \\
0 & \text { if } & x > \frac{\pi}{2}
\end{array}\right.
$$
Find the distribution function $F(x)$. | Solution. If $x \leq \pi / 4$, then $f(x)=0$, hence,
$$
F(x)=\int_{-\infty}^{x} f(x) d x=\int_{-\infty}^{x} 0 \cdot d x=0
$$
If $\pi / 4 < x \leq \pi / 2$, then
$$
F(x)=\int_{-\infty}^{\pi / 4} 0 \cdot d x+\int_{\pi / 4}^{x} 2 \sin 2 x d x=0-\left.\cos 2 x\right|_{\pi / 4} ^{x}=-\cos 2 x + \cos \frac{\pi}{2} = -\cos... | F(x)={\begin{pmatrix}0&\text{if}&x\leq\frac{\pi}{4}\\-\cos2x&\text{if}&\frac{\pi}{4}<x\leq\frac{\pi}{2}\\1&\text{if}&x>\frac{\pi}{2}\end{pmatrix}.} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,718 |
110. A random variable $X$, all possible values of which belong to the interval ( $0 ; \pi / 3$ ), is given in this interval by the probability density function $f(x)=C \cdot \sin 3 x$. Find the coefficient $C$. | Solution. If the function $f(x)$ represents the probability density function of a continuous random variable $X$, defined in the interval $(a, b)$, then the condition $\int_{a}^{b} f(x) d x=1$ is satisfied.
To determine the unknown coefficient $C$ in the expression of the function $f(x)=C \cdot \sin 3 x$ so that this ... | \frac{3}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,719 |
112. A random variable $X$ is subject to Simpson's law (the law of the isosceles triangle) on the interval $x \in[-c ; c]$ (Fig. 18). Find:
a) the probability density function of this random variable;
b) the probability of the variable $X$ falling into the interval $(c / 2, c)$. | The area bounded by the graph of the function $f(x)$, which characterizes the probability density function of a random variable $X$, and the $O x$ axis is equal to one. Since the graph of the probability density function of the random variable $X$ is shown in Fig. 18, the area of the triangle $A B C$ must be equal to o... | \frac{7}{8} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,721 |
118. The random variable $X$ is given by the distribution function:
$$
F(x)=\left\{\begin{array}{ccc}
0 & \text { if } & x \leq -c \\
\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c} & \text { if } & -c < x \leq c \\
1 & \text { if } & x > c
\end{array}\right.
$$
( arcsine law ).
Find the mathematical expectation of th... | Solution. Let's find the probability density function of the random variable $X: f(x)=F^{\prime}(x)$. For $x \leq -c$ and for $x > c$, the function $f(x)=0$. For $-c < x \leq c$, we have
$$
f(x)=\left(\frac{1}{2}+\frac{1}{\pi} \arcsin \frac{x}{c}\right)^{\prime}=\frac{1}{\pi \sqrt{c^{2}-x^{2}}}
$$
The expected value ... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,722 |
121. Find the mathematical expectation and variance of a random variable $X$ distributed uniformly for $x \in [a ; b]$. | S o l u t i o n. The expected value of a random variable $X$, uniformly distributed for $x \in[a ; b]$, will be found using formula (54), into which we will substitute the expression $f(x)$ given by formula (60):
$$
M(x)=\int_{a}^{b} x f(x) d x=\frac{1}{b-a} \int_{a}^{b} x d x=\frac{b^{2}-a^{2}}{2(b-a)}=\frac{a+b}{2}
... | M(X)=\frac{+b}{2},\quadD(X)=\frac{(b-)^{2}}{12} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,723 |
123. On the highway, there is an automatic traffic light that shows green light for 1 minute for vehicles and red light for 45 seconds, then again 1 minute of green light and 45 seconds of red light, and so on. A car passes along the highway at a random moment in time, unrelated to the operation of the traffic light. F... | S o l u t i o n. Let's consider the random variable $T$ - the moment of time when a car passes by the traffic light within an interval equal to the period of the traffic light color changes. The period of the traffic light color changes is $(1+3 / 4)=7 / 4$ minutes. The random variable $T$ is uniformly distributed on t... | \frac{4}{7} | Other | math-word-problem | Yes | Yes | olympiads | false | 31,724 |
126. A substance is being weighed without systematic errors. Random errors $X$ in weighing are subject to a normal distribution with a root mean square deviation $\sigma=20$ g. Find the probability that the weighing will be performed with an error not exceeding $10 \mathrm{r}$ in absolute value. | S o l u t i o n. The random variable $X$ - random weighing errors. The expected value of $X$ is $a=0$. The required probability will be found using formula (63) with $\varepsilon=10$:
$$
P(|X-0| \leq 10)=2 \Phi(10 / 20)=2 \Phi(0.5)
$$
From the table of values of the function $\Phi(x)$ (see table 3 in the appendix), w... | 0.383 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,725 |
127. The fleece weight of the Ovsiy breed sheep is a random variable $X$ distributed according to the normal law. Practically all possible values of this variable belong to the interval $(7 ; 10.6)$ kg. Find the interval, symmetric with respect to the mathematical expectation, in which with a probability of 0.95 the po... | S o l u t i o n. The random variable $X$ - the wool clip in Aksaniyska breed sheep. Practically all possible values of any random variable distributed according to the normal law are contained in the interval $a \pm 3 \sigma$, where $a$ and $\sigma$ are its mathematical expectation and standard deviation (the three-sig... | (7.624,9.976) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,726 |
130. The diameters of balls for bearings are controlled as follows: if a ball does not pass through a hole of diameter $d_{1}$, but does pass through a hole of diameter $d_{2}>d_{1}$, then its size is considered acceptable. If either of these conditions is not met, the ball is rejected. It is known that the diameter of... | S o l u t i o n. From the formula $a=\left(d_{1}+d_{2}\right) / 2$, it follows that the mathematical expectation $a$ of the considered random variable $X$ is the midpoint of the interval ( $d_{1}, d_{2}$ ), i.e., the sizes $d_{1}$ and $d_{2}$ are symmetric with respect to $a$. Considering that $\varepsilon=\left(d_{2}-... | 0.0456 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,727 |
133. The student remembers that the probability density function of the exponential distribution has the form $f(x)=0$ for $x<0$, and for $x \geq 0$
$f(x)=C e^{-\lambda x}$; however, he forgot the value of the constant $C$. Find the parameter $C$. | S o l u t i o n. To find $C$, we will use the property of the probability density function: $\int_{-\infty}^{\infty} f(x) d x=1$. For $x<0$, the function $f(x)=0$, and for $x \geq 0$, the function $f(x)=C e^{-\lambda x}$, so the condition must be satisfied: $\int_{0}^{\infty} C e^{-\lambda x} d x=1$.
We will find the ... | \lambda | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,729 |
135. Two independently operating elements are tested. The duration of failure-free operation of the first and second elements are random variables $T_{1}$ and $T_{2}$, distributed according to the exponential law; These variables are characterized by the distribution functions
$$
F_{1}(t)=P\left(T_{1}<t\right)=1-e^{-0... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,730 | ||
1. a) $(x+3 y)^{2}$;
б) $(2 x+3 y)^{2}$
в) $\left(m^{3}+n^{5}\right)^{2}$;
г) $(5 x+3 y)^{2}$
д) $\left(3 m^{5}-4 n^{2}\right)^{2}$. | Solution. a) $(x+3 y)^{2}=x^{2}+2(x \cdot 3 y)+(3 y)^{2}=x^{2}+6 x y+9 y^{2}$;
b) $(2 x+3 y)^{2}=(2 x)^{2}+2(2 x \cdot 3 y)+(3 y)^{2}=4 x^{2}+12 x y+9 y^{2}$;
c) $\left(m^{3}+n^{5}\right)^{2}=\left(m^{3}\right)^{2}+2\left(m^{3} n^{5}\right)+\left(n^{5}\right)^{2}=m^{6}+2 m^{3} n^{5}+n^{10}$.
Let's recall that when r... | )x^{2}+6xy+9y^{2};b)4x^{2}+12xy+9y^{2};)^{6}+2^{3}n^{5}+n^{10};)25x^{2}-30xy+9y^{2};e)9^{10}-24^{5}n^{2}+1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,731 |
11. a) $x^{2}+2 x+1$; 6) $m^{2}-4 m n+4 n^{2}$. | Solution. a) $x^{2}+2 x+1=(x)^{2}+2 \cdot x \cdot 1+1^{2}=(x+1)^{2}$;
b) $m^{2}-4 m n+4 n^{2}=(m)^{2}-2(m \cdot 2 n)+(2 n)^{2}=(m-2 n)^{2}$. | (x+1)^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,732 |
36. a) $(a+2 b)^{3}$;
б) $(5 a-b)^{3}$
в) $(2 a+3 b)^{3}$;
г) $\left(m^{3}-n^{2}\right)^{3}$. | Solution.
a) $(a+2 b)^{3}=a^{3}+3 a^{2}(2 b)+3 a(2 b)^{2}+(2 b)^{3}=a^{3}+6 a^{2} b+12 a b^{2}+8 b^{3}$;
b) $(5 a-b)^{3}=(5 a)^{3}+3(5 a)^{2}(-b)+3 \cdot 5 a(-b)^{2}+(-b)^{3}=125 a^{3}-$
$-75 a^{2} b+15 a b^{2}-b^{3}$
c) $(2 a+3 b)^{3}=(2 a)^{3}+3(2 a)^{2} \cdot 3 b+3 \cdot 2 a(3 b)^{2}+(3 b)^{3}=8 a^{3}+36 a^{2} b... | )^{3}+6^{2}b+12^{2}+8b^{3};b)125^{3}-75^{2}b+15^{2}-b^{3};)8^{3}+36^{2}b+54^{2}+27b^{3};)^{9}-3^{6} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,734 |
47. a) $(a+1)(a-1)$; b) $(2 a+3)(2 a-3)$; c) $\left(m^{3}-n^{5}\right)\left(n^{5}+m^{3}\right)$; d) $\left(3 m^{2}-5 n^{2}\right)\left(3 m^{2}+5 n^{2}\right)$. | Solution. a) $(a+1)(a-1)=a^{2}-1^{2}=a^{2}-1$;
b) $(2 a+3)(2 a-3)=(2 a)^{2}-3^{2}=4 a^{2}-9$;
c) $\left(m^{3}-n^{5}\right)\left(n^{5}+m^{3}\right)=\left(m^{3}\right)^{2}-\left(n^{5}\right)^{2}=m^{6}-n^{10}$;
d) $\left(3 m^{2}-5 n^{2}\right)\left(3 m^{2}+5 n^{2}\right)=\left(3 m^{2}\right)^{2}-\left(5 n^{2}\right)^{2}... | )^{2}-1;b)4^{2}-9;)^{6}-n^{10};)9^{4}-25n^{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,735 |
56. a) $29 \cdot 31$; b) $86^{2}-14^{2}$. | Solution. a) $29 \cdot 31=(30-1)(30+1)=30^{2}-1^{2}=900-1=899$;
b) $86^{2}-14^{2}=(86+14)(86-14)=100 \cdot 72=7200$. | 899 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,736 |
66. a) $(x+y)\left(x^{2}-x y+y^{2}\right)$; b) $(x+3)\left(x^{2}-3 x+9\right)$; c) $(x-1)\left(x^{2}+x+1\right)$; d) $(2 x-3)\left(4 x^{2}+6 x+9\right)$ | Solution. a) $(x+y)\left(x^{2}-x y+y^{2}\right)=x^{3}+y^{3}$;
b) $(x+3)\left(x^{2}-3 x+9\right)=x^{3}+27$
c) $(x-1)\left(x^{2}+x+1\right)=x^{3}-1$;
d) $(2 x-3)\left(4 x^{2}+6 x+9\right)=(2 x)^{3}-3^{3}=8 x^{3}-27$. | )x^{3}+y^{3};b)x^{3}+27;)x^{3}-1;)8x^{3}-27 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,737 |
77. Simplify the expression
$$
\frac{3 a^{2}+3 a b+3 b^{2}}{4 a+4 b} \cdot \frac{2 a^{2}-2 b^{2}}{9 a^{3}-9 b^{3}}
$$ | Solution. In the numerator and denominator of each fraction, we factor out the common factor:
$$
\frac{3 a^{2}+3 a b+3 b^{2}}{4 a+4 b} \cdot \frac{2 a^{2}-2 b^{2}}{9 a^{3}-9 b^{3}}=\frac{3\left(a^{2}+a b+b^{2}\right)}{4(a+b)} \cdot \frac{2\left(a^{2}-b^{2}\right)}{9\left(a^{3}-b^{3}\right)}
$$
Using the formulas for ... | \frac{1}{6} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,738 |
108. a) $2^{-3}$; b) $\left(\frac{1}{3}\right)^{-2}$; c) $\left(\frac{2}{3}\right)^{-4} ;$ d) $(-0.2)^{-3}$. | Solution. a) $2^{-3}=\frac{1}{2^{3}}=\frac{1}{8} ;$ b) $\left(\frac{1}{3}\right)^{-2}=\frac{1}{\left(\frac{1}{3}\right)^{2}}=9$;
c) $\left(\frac{2}{3}\right)^{-4}=\frac{1}{\left(\frac{2}{3}\right)^{4}}=\left(\frac{3}{2}\right)^{4}=\frac{81}{16}$;
d) $(-0.2)^{-3}=\frac{1}{(-0.2)^{3}}=\frac{1}{-0.008}=-125$. | \frac{1}{8},9,\frac{81}{16},-125 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,739 |
120. Calculate $\left(\frac{9}{16}\right)^{-1 / 10}:\left(\frac{25}{36}\right)^{-3 / 2}-\left[\left(\frac{4}{3}\right)^{-1 / 2}\right]^{-2 / 5}\left(\frac{6}{5}\right)^{-3}$. | Solution. We will perform the actions sequentially:
1) $\left(\frac{9}{16}\right)^{-1 / 10}=\left[\left(\frac{3}{4}\right)^{2}\right]^{-1 / 10}=\left(\frac{3}{4}\right)^{-1 / 5}=\left(\frac{4}{3}\right)^{1 / 5}$;
2) $\left(\frac{25}{36}\right)^{-3 / 2}=\left[\left(\frac{5}{6}\right)^{2}\right]^{-3 / 2}=\left(\frac{5}{... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,741 |
129. Simplify the fraction $\frac{x^{3 / 4}-25 x^{1 / 4}}{x^{1 / 2}+5 x^{1 / 4}}$. | Solution. By factoring the numerator and denominator of the fraction and simplifying it, we get
$$
\frac{x^{3 / 4}-25 x^{1 / 4}}{x^{1 / 2}+5 x^{1 / 4}}=\frac{x^{1 / 4}\left(x^{1 / 2}-25\right)}{x^{1 / 4}\left(x^{1 / 4}+5\right)}=\frac{\left(x^{1 / 4}-5\right)\left(x^{1 / 4}+5\right)}{\left(x^{1 / 4}+5\right)}=x^{1 / 4... | x^{1/4}-5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,742 |
137. $5^{x}=625$ | Solution. Writing 625 as $5^{4}$, we get $5^{x}=5^{4}$, hence $x=4$. 138. $8^{x}=32$.
Solution. We have $32=2^{5} ; 8^{x}=\left(2^{3}\right)^{x}=2^{3 x}$. Therefore, $2^{3 x}=2^{5}$, hence $3 x=5$, i.e., $x=5 / 3$. | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,743 |
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