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”.
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.
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.
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 \(|a|\geq \frac{1}{3}\).
On one island, one tribe has a custom – during the ritual dance, the leader throws up three thin straight rods of the same length, connected in the likeness of the letter capital \(\pi\), \(\Pi\). The adjacent rods are connected by a short thread and therefore freely rotate relative to each other. The bars fall on the sand, forming a random figure. If it turns out that there is self-intersection (the first and third bars cross), then the tribe in the coming year are waiting for crop failures and all sorts of trouble. If there is no self-intersection, then the year will be successful – satisfactory and happy. Find the probability that in 2019, the rods will predict luck.
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?
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?
Three cyclists travel in one direction along a circular track that is 300 meters long. Each of them moves with a constant speed, with all of their speeds being different. A photographer will be able to make a successful photograph of the cyclists, if all of them are on some part of the track which has a length of \(d\) meters. What is the smallest value of \(d\) for which the photographer will be able to make a successful photograph sooner or later?
We took several positive numbers and constructed the following sequence: \(a_1\) is the sum of the initial numbers, \(a_2\) is the sum of the squares of the original numbers, \(a_3\) is the sum of the cubes of the original numbers, and so on.
a) Could it happen that up to \(a_5\) the sequence decreases (\(a_1> a_2> a_3> a_4> a_5\)), and starting with \(a_5\) – it increases (\(a_5 < a_6 < a_7 <\dots\))?
b) Could it be the other way around: before \(a_5\) the sequence increases, and starting with \(a_5\) – decreases?
A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of \(n\) cells. We will call the natural number \(n\) jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one \(n > 50\) that is not jumpable.