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.
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 \(1 + 1/2 + 1/3 + \dots + 1/n\).
Are there functions \(p (x)\) and \(q (x)\) such that \(p (x)\) is an even function and \(p (q (x))\) is an odd function (different from identically zero)?
Author: A.K. Tolpygo
12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?
One hundred cubs found berries in the forest: the youngest managed to grab 1 berry, the next youngest cub – 2 berries, the next – 4 berries, and so on, until the oldest who got \(2^{99}\) berries. The fox suggested that they share the berries “fairly.” She can approach two cubs and distribute their berries evenly between them, and if this leaves an extra berry, then the fox eats it. With such actions, she continues, until all the cubs have an equal number of berries. What is the largest number of berries that the fox can eat?