Problems

\(N\) pairs of socks hang on a washing line in a random order (the order in which they were taken out of the washing machine). There are no two identical pairs. The socks hang under the drying sheet, so the Scattered Scientist takes out one toe by touch and compares each new sock with all of the previous ones. Find the mathematical expectation of the number of socks taken at the moment when the Scientist will have some pair.

An ant goes out of the origin along a line and makes \(a\) steps of one unit to the right, \(b\) steps of one unit to the left in some order, where \(a > b\). The wandering span of the ant is the difference between the largest and smallest coordinates of the ant for the entire length of its journey.

a) Find the largest possible wandering range.

b) Find the smallest possible range.

c) How many different sequences of motion of the ant are there, where the wandering range is the greatest possible?

We will assume that the birth of a girl and a boy is equally probable. It is known that in some family there are two children.

a) What is the probability that one of them is a boy and one a girl?

b) Additionally, it is known that one of the children is a boy. What is the probability that there is one boy and one girl in the family now?

c) Additionally, it is known that the boy was born on a Monday. What is the probability that there is one boy and one girl in the family now?

The figure shows a payment order to pay an electricity bill to some power supply company for March 2013.

Every month, the client sends the company the testimony of a three-tariff meter installed her the apartment. From the indications for the current month, the corresponding indications for the previous month are subtracted, and the actual monthly expenditure is obtained for each of the three tariff zones (peak, night, inter-peak). Then the expense for each zone is multiplied by the price of one kilowatt-hour in this zone. Adding the received amounts, the client receives the total amount of payment for a month. In this example, the customer will pay £660.72.

The company maintains a record of electricity consumption and payment, using the data received from the customer. The problem is that the company sometimes confuses the six numbers obtained, rearranging them in an arbitrary order, however, it ensures that the current reading remains greater than the previous one. As a result, the calculation of the company may be flawed. If the company believes that the client must pay more than she has paid, the company requires additional payment.

Using the data from the receipt shown, find:

a) the maximum possible amount of surcharge for March 2013, which the company will require from the client;

b) the mathematical expectation of the difference between the amount that the company calculates and the amount paid by the client.

Every day, Patrick the dog chews one slipper from the available stock in the house. Strictly with a probability of 0.5 Patrick wants to chew the left slipper, and with a probability of 0.5 – the right one. If the desired slippers are not present, Patrick becomes upset. How many pairs of the same slippers need to be bought, so that with a probability of not less than 0.8 Patrick does not get upset for an entire week (7 days)?

Find the probability that heads will fall an even number of times, in an experiment in which:

a) a symmetrical coin is thrown \(n\) times;

b) a coin is thrown \(n\) times, for which the probability of getting heads in one throw is \(p(0 < p < 1)\).

In Anchuria, presidential elections are being prepared, in which President Miraflores wants to win. Exactly half of the voters support Miraflores, and the other half support Dick Maloney. Miraflores is also a voter. According to the law, he has the right to divide all of the voters into two constituencies at his own discretion. In each of the districts, the voting is conducted as follows: each voter marks the name of their candidate on the ballot; all ballots are placed in the ballot box. Then one random ballot is chosen from the ballot box, and the one whose name is marked on it will win in this district. The candidate wins the election only if he wins in both districts. If the winner does not appear, the next round of voting is appointed according to the same rules. How should Miraflores divide the electorate in order to maximize the probability of his victory on the first round?

In a terrible thunderstorm, along a rope ladder, \(n\) dwarfs ascend in a chain. If suddenly there is a thunderbolt, then from fear, every gnome, regardless of others, can fall with probability \(p\) (\(0 < p < 1\)). If the dwarf falls, then he also hits all of the dwarfs that are below him. Find:

a) The probability that exactly \(k\) dwarfs will fall.

b) The mathematical expectation of the number of fallen dwarfs.

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.