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On a lottery ticket, it is necessary for Mary to mark 8 cells from 64. What is the probability that after the draw, in which 8 cells from 64 will also be selected $($all such possibilities are equally probable$)$, it turns out that Mary guessed

a) exactly 4 cells? b) exactly 5 cells? c) all 8 cells?

According to one implausible legend, Cauchy and Bunyakovsky were very fond of playing darts in the evenings. But the target was unusual – the sectors on it were unequal, so the probability of getting into different sectors was not the same. Once Cauchy throws a dart and hits the target. Bunyakovsky throws the next one. Which is more likely: that Bunyakovsky will hit the same sector that Cauchy’s dart went into, or that his dart will land on the next sector clockwise?

An incredible legend says that one day Stirling was considering the numbers of Stirling of the second kind. During his thoughtfulness, he threw 10 regular dice on the table. After the next throw, he suddenly noticed that in the dropped combination of points there were all of the numbers from 1 to 6. Immediately Stirling reflected: what is the probability of such an event? What is the probability that when throwing 10 dice each number of points from 1 to 6 will drop out on at least one die?

There are $n$ random vectors of the form $(y_1, y_2, y_3)$, where exactly one random coordinate is equal to 1, and the others are equal to 0. They are summed up. A random vector a with coordinates $(Y_1, Y_2, Y_3)$ is obtained.

a) Find the mathematical expectation of a random variable $a^2$.

b) Prove that $|a|\geq \frac{1}{3}$.

A sequence consists of 19 ones and 49 zeros, arranged in a random order. We call the maximal subsequence of the same symbols a “group”. For example, in the sequence 110001001111 there are five groups: two ones, then three zeros, then one one, then two zeros and finally four ones. Find the mathematical expectation of the length of the first group.

On weekdays, the Scattered Scientist goes to work along the circle line on the London Underground from Cannon Street station to Edgware Road station, and in the evening he goes back $($see the diagram$)$.

Entering the station, the Scientist sits down on the first train that arrives. It is known that in both directions the trains run at approximately equal intervals, and along the northern route $($via Farringdon$)$ the train goes from Cannon Street to Edgware Road or back in 17 minutes, and along the southern route $($via St James Park$)$ – 11 minutes. According to an old habit, the scientist always calculates everything. Once he calculated that, from many years of observation:

– the train going counter-clockwise, comes to Edgware Road on average 1 minute 15 seconds after the train going clockwise arrives. The same is true for Cannon Street.

– on a trip from home to work the Scientist spends an average of 1 minute less time than a trip home from work.

Find the mathematical expectation of the interval between trains going in one direction.

At the ball, there were n married couples. In each pair, the husband and wife are of the same height, but there are no two pairs of the same height. The waltz begins, and all those who came to the ball randomly divide into pairs: each gentleman dances with a randomly chosen lady.

Find the mathematical expectation of the random variable X, “the number of gentlemen who are shorter than their partners”.

In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals $($it could be that some pairs will repeat$)$. The prize is given to those who win both matches. Find:

a) the smallest possible number of tournament winners;

b) the mathematical expectation of the number of tournament winners.

At the Antarctic station, there are n polar explorers, all of different ages. With the probability p between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.

The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions $($or sectors$)$, numbered from 1 to 13, that are thought up and sent in by the viewers of the programme. Envelopes with the questions are selected in turn in random order with the help of a spinning top with an arrow. If this sector has already come up previously, and the envelope is no longer there, then the next clockwise sector is played. If it is also empty, then the next one is played, etc., until there is a non-empty sector.

Before the break, the players played six sectors.

a) What is more likely: that sector number 1 has already been played or that sector number 8 has already been plated?

b) Find the probability that, before the break, six sectors with numbers from 1 to 6 were played consecutively.

When one scientist comes up with an ingenious idea, he writes it down on a piece of paper, but then he realises that the idea is not brilliant, scrunches up this sheet of paper and throws it under the table, where there are two rubbish bins. The scientist misses the first bin with a probability p $>$ 0.5, and with the same probability he misses the second. In the morning, the scientist threw five crumpled brilliant ideas under the table. Find the probability that there was at least one of these ideas in each bin.

In a shopping centre, three machines sell coffee. During the day, the first machine can break down with a probability of 0.4 and the second with a probability of 0.3. Every evening, Mr Ivanov, the mechanic, comes and repairs all of the broken-down coffee machines. One day, Ivanov wrote, in his report, that the mathematical expectation of breakdowns during one week is 12. Prove that Mr Ivanov is exaggerating.

The teacher on probability theory leaned back in his chair and looked at the screen. The list of those who signed up is ready. The total number of people turned out to be n. Only they are not in alphabetical order, but in a random order in which they came to the class.

“We need to sort them alphabetically,” the teacher thought, “I’ll go down in order from the top down, and if necessary I’ll rearrange the student’s name up in a suitable place. Each name should be rearranged no more than once”.

Prove that the mathematical expectation of the number of surnames that you do not have to rearrange is 1 + 1/2 + 1/3 + … + 1/n.

A toy cube is symmetrical, but it’s unusual: two faces have two points, and the other four have one point. Sarah threw the cube several times, and as a result, the sum of all of the points was 3. Find the probability that one throw resulted in the face with 2 points coming up.

Chess board fields are numbered in rows from top to bottom by the numbers from 1 to 64. 6 rooks are randomly assigned to the board, which do not capture each other $($one of the possible arrangements is shown in the figure$)$. Find the mathematical expectation of the sum of the numbers of fields occupied by the rooks.

A ticket for a train costs 50 pence, and the penalty for a ticketless trip is 450 pence. If the free rider is discovered by the controller, he pays both the penalty and the ticket price. It is known that the controller finds the free rider on average once out of every 10 trips. The free rider got acquainted with the basics of probability theory and decided to adhere to a strategy that gives the mathematical expectation of spending the smallest possible. How should he act: buy a ticket every time, never buy one, or throw a coin to determine whether he should buy a ticket or not?

A table of size $3 \times 3$ $($as for playing tic-tac-toe$)$ is given. Four chips are put $($one each$)$ on four randomly selected cells. Find the probability that among these four chips there are three that stand in a row vertically, horizontally or diagonally.

Two hockey teams of the same strength agreed that they will play until the total score reaches 10.

Find the mathematical expectation of the number of times when there is a draw.

Every Friday ten gentlemen come to the club, and each one gives the doorman his hat. Each hat is just right for its owner, but there are no two identical hats. The gentlemen leave one by one in a random order.

Seeing off the next gentleman, the club’s doorman tries to put the first hat that he grabs on the gentleman’s head. If it fits $($not necessarily perfectly$)$, the gentleman leaves with that hat. If it is too small, the doorman tries the next random hat from the remaining ones. If all of the remaining hats turned out to be too small, the doorman says to the poor fellow: “Sir, you do not have a hat today,” and the gentleman goes home with his head uncovered. Find the probability that next Friday the doorman will not have a single hat.

A high rectangle of width 2 is open from above, and the L-shaped domino falls inside it in a random way $($see the figure$)$.

a) $k$ $L$-shaped dominoes have fallen. Find the mathematical expectation of the height of the resulting polygon.

b) $7$ $G$-shaped dominoes fell inside the rectangle. Find the probability that the resulting figure will have a height of 12.