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.
At a factory known to us, we cut out metal disks with a diameter of 1 m. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, a measurement error occurs, and therefore the standard deviation of the radius is 10 mm. Engineer Gavin believes that a stack of 100 disks on average will weigh 10,000 kg. By how much is the engineer Gavin wrong?
At a conference there were 18 scientists, of which exactly 10 know the eye-popping news. During the break (coffee break), all scientists are broken up into random pairs, and in each pair, anyone who knows the news, tells this news to another if he did not already know it.
a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.
b) Find the probability that after the coffee break the number of scientists who know the news will be 14.
c) Denote by the letter \(X\) the number of scientists who know the eye-popping news after the coffee break. Find the mathematical expectation of \(X\).
Two hockey teams of the same strength agreed that they will play until the total score reaches 10. Find the mathematical expectation of the number of times when there is a draw.
A ticket for a train costs 50 pence, and the penalty for a ticketless trip is 450 pence. If the free rider is discovered by the controller, he pays both the penalty and the ticket price. It is known that the controller finds the free rider on average once out of every 10 trips. The free rider got acquainted with the basics of probability theory and decided to adhere to a strategy that gives the mathematical expectation of spending the smallest possible. How should he act: buy a ticket every time, never buy one, or throw a coin to determine whether he should buy a ticket or not?
Chess board fields are numbered in rows from top to bottom by the numbers from 1 to 64. 6 rooks are randomly assigned to the board, which do not capture each other (one of the possible arrangements is shown in the figure). Find the mathematical expectation of the sum of the numbers of fields occupied by the rooks.
The teacher on probability theory leaned back in his chair and looked at the screen. The list of those who signed up is ready. The total number of people turned out to be \(n\). Only they are not in alphabetical order, but in a random order in which they came to the class.
“We need to sort them alphabetically,” the teacher thought, “I’ll go down in order from the top down, and if necessary I’ll rearrange the student’s name up in a suitable place. Each name should be rearranged no more than once”.
Prove that the mathematical expectation of the number of surnames that you do not have to rearrange is \(1 + 1/2 + 1/3 + \dots + 1/n\).
In a shopping centre, three machines sell coffee. During the day, the first machine can break down with a probability of 0.4 and the second with a probability of 0.3. Every evening, Mr Ivanov, the mechanic, comes and repairs all of the broken-down coffee machines. One day, Ivanov wrote, in his report, that the mathematical expectation of breakdowns during one week is 12. Prove that Mr Ivanov is exaggerating.
At the Antarctic station, there are \(n\) polar explorers, all of different ages. With the probability \(p\) between each two polar explorers, friendly relations are established, regardless of other sympathies or antipathies. When the winter season ends and it’s time to go home, in each pair of friends the senior gives the younger friend some advice. Find the mathematical expectation of the number of those who did not receive any advice.
In a tournament, 100 wrestlers are taking part, all of whom have different strengths. In any fight between two wrestlers, the one who is stronger always wins. In the first round the wrestlers broke into random pairs and fought each other. For the second round, the wrestlers once again broke into random pairs of rivals (it could be that some pairs will repeat). The prize is given to those who win both matches. Find:
a) the smallest possible number of tournament winners;
b) the mathematical expectation of the number of tournament winners.