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.

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.

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.

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 the power plant, rectangles that are 2 m long and 1 m wide are produced. The length of the objects is measured by the worker Howard, and the width, irrespective of Howard, is measured by the worker Rachel. The average error is zero for both, but Howard allows a standard measurement error (standard deviation of length) of 3 mm, and Rachel allows a standard error of 2 mm.

a) Find the mathematical expectation of the area of the resulting rectangle.

b) Find the standard deviation of the area of the resulting rectangle in centimetres squared.