We throw a symmetrical coin \(n\) times. Suppose that heads came up \(m\) times. The number \(m/n\) is called the frequency of the fall of heads. The number \(m/n - 0.5\) is called the frequency deviation from the probability, and the number \(|m/n - 0.5|\) is called the absolute deviation. Note that the deviation and the absolute deviation are random variables. For example, if a coin was thrown 5 times and heads came up two times, the deviation is equal to \(2/5 - 0.5 = -0.1\), and the absolute deviation is 0.1.
The experiment consists of two parts: first the coin is thrown 10 times, and then – 100 times. In which of these cases is the mathematical expectation of the absolute deviation of the frequency of getting heads is greater than the probability?
In the magical land of Anchuria there are only \(K\) laws and \(N\) ministers. The probability that a randomly chosen minister knows a randomly chosen law is \(p\). One day, the ministers gathered for a meeting, to write the Constitution. If at least one minister knows the law, then this law will be taken into account in the Constitution, otherwise this law will not be taken into account in the Constitution. Find:
a) The probability that exactly \(M\) laws will be taken into account into the Constitution.
b) The mathematical expectation of the number of registered laws.
There is a deck of playing cards on the table (for example, in a row). On top of each card we put a card from another deck. Some cards may have coincided. Find:
a) the mathematical expectation of the number of cards that coincide;
b) the variance of the number of cards that coincide.
The tower in the castle of King Arthur is crowned with a roof, which is a triangular pyramid, in which all flat angles at the top are straight. Three roof slopes are painted in different colours. The red roof slope is inclined to the horizontal at an angle \(\alpha\), and the blue one at an angle \(\beta\). Find the probability that a raindrop that fell vertically on the roof in a random place fell on the green area.
If one person spends one minute waiting, we will say that one human-minute is spent aimlessly. In the queue at the bank, there are eight people, of which five plan to carry out simple operations, which take 1 minute, and the others plan to carry out long operations, taking 5 minutes. Find:
a) the smallest and largest possible total number of aimlessly spent human-minutes;
b) the mathematical expectation of the number of aimlessly spent human-minutes, provided that customers queue up in a random order.
There are 9 street lamps along the road. If one of them does not work but the two next to it are still working, then the road service team is not worried about it. But if two lamps in a row do not work then the road service team immediately changes all non-working lamps. Each lamp does not work independently of the others.
a) Find the probability that the next replacement will include changing 4 lights.
b) Find the mathematical expectation of the number of lamps that will have to be changed on the next replacement.
What is the smallest number of cells that can be chosen on a \(15\times15\) board so that a mouse positioned on any cell on the board touches at least two marked cells? (The mouse also touches the cell on which it stands.)
What is the largest number of horses that can be placed on an \(8\times8\) chessboard so that no horse touches more than seven of the others?
Harry thought of two positive numbers \(x\) and \(y\). He wrote down the numbers \(x + y\), \(x - y\), \(xy\) and \(x/y\) on a board and showed them to Sam, but did not say which number corresponded to which operation.
Prove that Sam can uniquely figure out \(x\) and \(y\).
At a contest named “Ah well, monsters!”, 15 dragons stand in a row. Between neighbouring dragons the number of heads differs by 1. If the dragon has more heads than both of his two neighbors, he is considered cunning, if he has less than both of his neighbors – strong, the rest (including those standing at the edges) are considered ordinary. In the row there are exactly four cunning dragons – with 4, 6, 7 and 7 heads and exactly three strong ones – with 3, 3 and 6 heads. The first and last dragons have the same number of heads.
a) Give an example of how this could occur.
b) Prove that the number of heads of the first dragon in all potential examples is the same.