Gary drew an empty table of \(50 \times 50\) and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row (the “multiplication table”). What is the largest number of products in this table which could be rational numbers?
There is an elastic band and glass beads: four identical red ones, two identical blue ones and two identical green ones. It is necessary to string all eight beads on the elastic band in order to get a bracelet. How many different bracelets can be made so that beads of the same colour are not next to each other? (Assume that there is no buckle, and the knot on the elastic is invisible).
To test a new program, a computer selects a random real number \(A\) from the interval \([1, 2]\) and makes the program solve the equation \(3x + A = 0\). Find the probability that the root of this equation is less than \(0.4\).
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.
The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions (or sectors), numbered from 1 to 13, that are thought up and sent in by the viewers of the programme. Envelopes with the questions are selected in turn in random order with the help of a spinning top with an arrow. If this sector has already come up previously, and the envelope is no longer there, then the next clockwise sector is played. If it is also empty, then the next one is played, etc., until there is a non-empty sector.
Before the break, the players played six sectors.
a) What is more likely: that sector number 1 has already been played or that sector number 8 has already been plated?
b) Find the probability that, before the break, six sectors with numbers from 1 to 6 were played consecutively.
On the school board a chairman is chosen. There are four candidates: \(A\), \(B\), \(C\) and \(D\). A special procedure is proposed – each member of the council writes down on a special sheet of candidates the order of his preferences. For example, the sequence \(ACDB\) means that the councilor puts \(A\) in the first place, does not object very much to \(C\), and believes that he is better than \(D\), but least of all would like to see \(B\). Being placed in first place gives the candidate 3 points, the second – 2 points, the third – 1 point, and the fourth - 0 points. After collecting all the sheets, the election commission summarizes the points for each candidate. The winner is the one who has the most points.
After the vote, \(C\) (who scored fewer points than everyone) withdrew his candidacy in connection with his transition to another school. They did not vote again, but simply crossed out \(B\) from all the leaflets. In each sheet there are three candidates left. Therefore, first place was worth 2 points, the second – 1 point, and the third – 0 points. The points were summed up anew.
Could it be that the candidate who previously had the most points, after the self-withdrawal of \(B\) received the fewest points?
In one box, there are two pies with mushrooms, in another box there are two with cherries and in the third one, there is one with mushrooms and one with cherries. The pies look and weigh the same, so it’s not known what is in each one. The grandson needs to take one pie to school. The grandmother wants to give him a pie with cherries, but she is confused herself and can only determine the filling by breaking the pie, but the grandson does not want a broken pie, he wants a whole one.
a) Show that the grandmother can act so that the probability of giving the grandson a whole pie with cherries will be equal to \(2/3\).
b) Is there a strategy in which the probability of giving the grandson a whole pie with cherries is higher than \(2/3\)?
There were 50 white and black crows sitting on a birch, and the number of black crows was not less than the number of whites. On the oak, there too were white and black crows, and there were 50 of them in total. On the oak, the number of black crows was also not less than the number of white ones. It could be that there was the same number of black and white crows, or maybe even there was one black crow less than white crows. One random crow flew from the birch to the oak, and after a while another random crow (maybe the same one) flew from the oak to the birch. Which is more probable: that the number of white crows on the birch is the same as it was at first, or that it has changed?
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.