In a convex polygon, which has an odd number of vertices equal to \(2n + 1\), two independently of each other random diagonals are chosen. Find the probability that these diagonals intersect inside the polygon.
At a conference there were 18 scientists, of which exactly 10 know the eye-popping news. During the break (coffee break), all scientists are broken up into random pairs, and in each pair, anyone who knows the news, tells this news to another if he did not already know it.
a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.
b) Find the probability that after the coffee break the number of scientists who know the news will be 14.
c) Denote by the letter \(X\) the number of scientists who know the eye-popping news after the coffee break. Find the mathematical expectation of \(X\).
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.
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 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.
One day in autumn the Scattered Scientist glanced at his ancient wall clock and saw that three flies fell asleep on the dial. The first one slept exactly at the 12 o’clock mark on the clock, and the other two just as neatly settled on the marks of 2 hours and 5 hours. The scientist made measurements and determined that the hour hand does not threaten the flies, but the minute one will sweep them all in turn. Find the probability that exactly 40 minutes after the Scientist noticed the flies, exactly two flies out of three were swept away by the minute hand.
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?
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 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.
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 \(1 + 1/2 + 1/3 + \dots + 1/n\).