The point \(O\), lying inside the triangle \(ABC\), is connected by segments with the vertices of the triangle. Prove that the variance of the set of angles \(AOB\), \(AOC\) and \(BOC\) is less than a) \(10\pi ^2/27\); b) \(2\pi ^2/9\).
At the sound of the whistle of the PE teacher, all 10 boys and 7 girls lined up randomly.
Find the mathematical expectation of the value “the number of girls standing to the left of all of the boys.”
Hercules meets the three-headed snake, the Lernaean Hydra and the battle begins. Every minute, Hercules cuts one of the snake’s heads off. With probability \(\frac 14\) in the place of the chopped off head grows two new ones, with a probability of \(1/3\), only one new head will grow and with a probability of \(5/12\), not a single head will appear. The serpent is considered defeated if he does not have a single head left. Find the probability that sooner or later Hercules will beat the snake.
In the set \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), replace one number with two other integers so that the set variance and its mean remain unchanged.
Prince Charming, and another 49 men and 50 women are randomly seated around a round table. Let’s call a man satisfied, if a woman is sitting next to him. Find:
a) the probability that Prince Charming is satisfied;
b) the mathematical expectation of the number of satisfied men.
Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.
a) What is the fourth number written on the board?
b) What is the eighth number written on the board?
A cube is created from 27 playing blocks.
a) Find the probability that there are exactly 25 sixes on the surface of the cube.
b) Find the probability that there is at least one 1 on the surface of the cube.
c) Find the mathematical expectation of the number of sixes on the surface of the cube.
d) Find the mathematical expectation of the sum of the numbers that are on the surface of the cube.
e) Find the mathematical expectation of a random variable: “The number of different digits that are on the surface of the cube.”
The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions that are thought up and sent in by the viewers of the program. Envelopes with the questions are selected in turn in a random order with the help of a spinning top with an arrow. If the experts answer correctly, they earn a point, and if they answer incorrectly, the viewers get one point. The game ends as soon as one of the teams scored 6 points. Suppose that the abilities of the teams of experts and viewers are equal.
a) Find the mathematical expectation of the number of points scored by the team of experts in 100 games.
b) Find the probability that, in the next game, envelope number 5 will come up.
On board the airplane there are \(2n\) passengers, and the airline loaded for them \(n\) servings of lunch with chicken and \(n\) servings with fish. It is known that a passenger with a probability of 0.5 prefers chicken and with a probability of 0.5 prefers fish. Let’s call a passenger dissatisfied if he does not have what he prefers.
a) Find the most likely number of dissatisfied passengers.
b) Find the mathematical expectation of the number of dissatisfied passengers.
c) Find the variance of the number of dissatisfied passengers.
In Anchuria, there is a single state examination. The probability of guessing the correct answer to each exam question is 0.25. In 2011, in order to obtain a certificate, it was necessary to answer correctly to 3 questions out of 20. In 2012, the Anchuria School of Management decided that 3 questions were not enough. Now you need to correctly answer 6 questions out of 40. It is asked, if you do not know anything but just guess the answers, in what year is the probability of obtaining an Anchurian certificate higher: in 2011 or 2012?