Problems
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.
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.)
a) There are three identical large vessels. In one there are 3 litres of syrup, in the other – 20 litres of water, and the third is empty. You can pour all the liquid from one vessel into another or into a sink. You can choose two vessels and pour into one of them liquid from the third, until the liquid levels in the selected vessels are equal. How can you get 10 litres of diluted 30% syrup?
b) The same, but there is \(N\) l of water. At what integer values of \(N\) can you get 10 liters of diluted 30% syrup?
Monica is in a broken space buggy at a distance of 18 km from the Lunar base, in which Rachel sits. There is a stable radio communication system between them. The air reserve in the space buggy is enough for 3 hours, in addition, Monica has an air cylinder for the spacesuit, with an air reserve of 1 hour. Rachel has a lot of cylinders with an air supply of 2 hours each. Rachel can not carry more than two cylinders at the same time (one of them she uses herself). The speed of movement on the Moon in the suit is 6 km/h. Could Rachel save Monica and not die herself?