Problems

In the magical land of Anchuria there are only $K$ laws and $N$ ministers. The probability that a randomly chosen minister knows a randomly chosen law is $p$. One day, the ministers gathered for a meeting, to write the Constitution. If at least one minister knows the law, then this law will be taken into account in the Constitution, otherwise this law will not be taken into account in the Constitution. Find:

a) The probability that exactly $M$ laws will be taken into account into the Constitution.

b) The mathematical expectation of the number of registered laws.

The probability of the birth of twins in Cambria is $p$, and no triplets are born in Cambria.

a) Evaluate the probability that a random Cambrian that one meets on the street is one of a pair of twins?

b) There are three children in a random Cambrian family. What is the probability that among them there is a pair of twins?

c) In Cambrian schools, twins must be enrolled in the same class. In total, there are $N$ first-graders in Cambria.

What is the expectation of the number of pairs of twins among them?

There is a deck of playing cards on the table (for example, in a row). On top of each card we put a card from another deck. Some cards may have coincided. Find:

a) the mathematical expectation of the number of cards that coincide;

b) the variance of the number of cards that coincide.

The tower in the castle of King Arthur is crowned with a roof, which is a triangular pyramid, in which all flat angles at the top are straight. Three roof slopes are painted in different colours. The red roof slope is inclined to the horizontal at an angle $\alpha$, and the blue one at an angle $\beta$. Find the probability that a raindrop that fell vertically on the roof in a random place fell on the green area.

If one person spends one minute waiting, we will say that one human-minute is spent aimlessly. In the queue at the bank, there are eight people, of which five plan to carry out simple operations, which take 1 minute, and the others plan to carry out long operations, taking 5 minutes. Find:

a) the smallest and largest possible total number of aimlessly spent human-minutes;

b) the mathematical expectation of the number of aimlessly spent human-minutes, provided that customers queue up in a random order.

There are 9 street lamps along the road. If one of them does not work but the two next to it are still working, then the road service team is not worried about it. But if two lamps in a row do not work then the road service team immediately changes all non-working lamps. Each lamp does not work independently of the others.

a) Find the probability that the next replacement will include changing 4 lights.

b) Find the mathematical expectation of the number of lamps that will have to be changed on the next replacement.

What is the minimum number of $1\times 1$ squares that need to be drawn in order to get an image of a $25\times 25$ square divided into 625 smaller 1x1 squares?

A box contains 100 red, blue, and white marbles. It is known that if we remove 26 marbles from the box, without looking, we will always have removed at least 10 marbles of one colour. What is the minimum number of marbles we need to remove to guarantee that we have removed 30 marbles of the same colour?

What is the largest number of horses that can be placed on an $8\times 8$ chessboard so that no horse touches more than seven of the others?

Harry thought of two positive numbers $x$ and $y$. He wrote down the numbers $x+y$, $x-y$, $xy$ and $x/y$ on a board and showed them to Sam, but did not say which number corresponded to which operation.

Prove that Sam can uniquely figure out $x$ and $y$.