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.
For the anniversary of the London Mathematical Olympiad, the mint coined three commemorative coins. One coin turned out correctly, the second coin on both sides had two heads, and the third had tails on both sides. The director of the mint, without looking, chose one of these three coins and tossed it at random. She got heads. What is the probability that the second side of this coin also has heads?
In a convex hexagon, independently of each other, two random diagonals are chosen. Find the probability that these diagonals intersect inside the hexagon (inside – that is, not at the vertex).
The shooter shoots at 3 targets until he shoots everything. The probability of a hit with one shot is \(p\).
a) Find the probability that he needs exactly 5 shots.
b) Find the mathematical expectation of the number of shots.
Ten tennis players came to the competitions, 4 of them were from Russia. According to the rules for the first round, the tennis players are broken into pairs randomly. Find the probability that in the first round, all Russian tennis players will play only with other Russian tennis players.
In the triangle \(ABC\), the angle \(A\) is equal to \(40^{\circ}\). The triangle is randomly thrown onto a table. Find the probability that the vertex \(A\) lies east of the other two vertices.
At the power plant, rectangles that are 2 m long and 1 m wide are produced. The length of the objects is measured by the worker Howard, and the width, irrespective of Howard, is measured by the worker Rachel. The average error is zero for both, but Howard allows a standard measurement error (standard deviation of length) of 3 mm, and Rachel allows a standard error of 2 mm.
a) Find the mathematical expectation of the area of the resulting rectangle.
b) Find the standard deviation of the area of the resulting rectangle in centimetres squared.