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.
The probability of the birth of twins in Cambria is \(p\), and no triplets are born in Cambria.
a) Evaluate the probability that a random Cambrian that one meets on the street is one of a pair of twins?
b) There are three children in a random Cambrian family. What is the probability that among them there is a pair of twins?
c) In Cambrian schools, twins must be enrolled in the same class. In total, there are \(N\) first-graders in Cambria.
What is the expectation of the number of pairs of twins among them?
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.
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.
Solve the equation \(2 \sin \pi x / 2 - 2 \cos \pi x = x^5 + 10x - 54\).
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 \(y\).
The functions \(f\) and \(g\) are defined on the entire number line and are reciprocal. It is known that \(f\) is represented as a sum of a linear and a periodic function: \(f (x) = kx + h (x)\), where \(k\) is a number, and \(h\) is a periodic function. Prove that \(g\) is also represented in this form.
It is known that \(a > 1\). Is it always true that \(\lfloor \sqrt{\lfloor \sqrt{a}\rfloor }\rfloor = \lfloor \sqrt{4}{a}\rfloor\)?