A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.
In the magical land of Anchuria there is a drafts championship made up of several rounds. The days and cities in which the rounds are carried out are determined by a draw. According to the rules of the championship, no two rounds can take place in one city, and no two rounds can take place on one day. Among the fans, a lottery is arranged: the main prize is given to those who correctly guess, before the start of the championship, in which cities and on which days all of the round will take place. If no one guesses, then the main prize will go to the organising committee of the championship. In total, there are eight cities in Anchuria, and the championship is only allotted eight days. How many rounds should there be in the championship, so that the organising committee is most likely to receive the main prize?
On a calculator keypad, there are the numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any keys. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last click.
a) The button with the multiplier sign breaks and does not work. The Scattered Scientist pressed several buttons in a random sequence. Which result of the resulting sequence of actions is more likely: an even number or an odd number?
b) Solve the previous problem if the multiplication symbol button is repaired.
A regular dice is thrown many times. Find the mathematical expectation of the number of rolls made before the moment when the sum of all rolled points reaches 2010 (that is, it became no less than 2010).
A fair dice is thrown many times. It is known that at some point the total amount of points became equal to exactly 2010.
Find the mathematical expectation of the number of throws made to this point.
Hercules meets the three-headed snake, the Lernaean Hydra and the battle begins. Every minute, Hercules cuts one of the snake’s heads off. With probability \(\frac 14\) in the place of the chopped off head grows two new ones, with a probability of \(1/3\), only one new head will grow and with a probability of \(5/12\), not a single head will appear. The serpent is considered defeated if he does not have a single head left. Find the probability that sooner or later Hercules will beat the snake.
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.
At the Antarctic station, there are \(n\) polar explorers, all of different ages. With the probability \(p\) between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.
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”.
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.