A coin is thrown 10 times. Find the probability that it never lands on two heads in a row.
\(N\) people lined up behind each other. The taller people obstruct the shorter ones, and they cannot be seen.
What is the mathematical expectation of the number of people that can be seen?
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?
The building has \(n\) floors and two staircases running from the first to the last floor. On each staircase between each two floors on the intermediate staircase there is a door separating the floors (it is possible to pass from the stairs to the floor, even if the door is locked). The porter decided that too many open doors is bad, and locked up exactly half of the doors, choosing the doors at random. What is the probability that you can climb from the first floor to the last, passing only through open doors?
In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.
A die is thrown six times. Find the mathematical expectation of the number of different faces the die lands on.
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.
On a Christmas tree, 100 light bulbs hang in a row. Then the light bulbs begin to switch according to the following algorithm: all are lit up, then after a second, every second light goes out, after another second, every third light bulb changes: if it was on, it goes out and vice versa. After another second, every fourth bulb switches, a second later – every fifth and so on. After 100 seconds the sequence ends. Find the probability that a light bulb straight after a randomly selected light bulb is on (bulbs do not burn out and do not break).
In the final of a contest of performances on March 8 two performances were left. In the first, \(n\) pupils from the class 5A performed, and in the second – \(n\) pupils of class 5B. At the play there were \(2n\) mothers of all \(2n\) students. The best performance is chosen by a vote by the mums. It is known that every mother, with a probability of \(\frac 12\), votes for the best performance and with a probability of \(\frac 12\) – for the performance in which her child participates.
a) Find the probability that the best performance will win with a majority of votes.
b) The same question, if more than two classes have reached the finals.
The mathematics teacher suggested changing the voting scheme at the performance competition. Currently, two groups compete in the final. In the first group, there are \(n\) pupils from class 5A, and in the second, there are \(n\) pupils from class 5B. \(2n\) mothers of all \(2n\) students attended the final of the competition. The best performance is chosen by the mothers voting. It is known that exactly half of the mothers vote honestly for the best performance, and the other half, in any case, vote for the performance in which her child participates (see problem number 65299). The teacher believes that it is necessary to choose a jury of \(2m\) people \((2m \ensuremath{\leq} n)\) from all \(2n\) mums at random. Find the probability that the best performance will win under such voting conditions.