In a school football tournament, 8 teams participate, each of which plays equally well in football. Each game ends with the victory of one of the teams. A randomly chosen by a draw number determines the position of the teams in the table:
What is the probability that teams \(A\) and \(B\):
a) will meet in the semifinals;
b) will meet in the finals.
Peter plays a computer game “A bunch of stones.” First in his pile of stones he has 16 stones. Players take turns taking from the pile either 1, 2, 3 or 4 stones. The one who takes the last stone wins. Peter plays this for the first time and therefore each time he takes a random number of stones, whilst not violating the rules of the game. The computer plays according to the following algorithm: on each turn, it takes the number of stones that leaves it to be in the most favorable position. The game always begins with Peter. How likely is it that Peter will win?
There are two symmetrical cubes. Is it possible to write some numbers on their faces so that the sum of the points when throwing these cubes on the upwards facing face on landing takes the values 1, 2, ..., 36 with equal probabilities?
In the first term of the year Daniel received five grades in mathematics with each of them being on a scale of 1 to 5, and the most common grade among them was a 5 . In this case it turned out that the median of all his grades was 4, and the arithmetic mean was 3.8. What grades could Daniel have?
There are fewer than 30 people in a class. The probability that at random a selected girl is an excellent student is \(3/13\), and the probability that at random a chosen boy is an excellent pupil is \(4/11\). How many excellent students are there in the class?
\(A\) and \(B\) shoot in a shooting gallery, but they only have one six-shot revolver with one cartridge. Therefore, they agreed in turn to randomly rotate the drum and shoot. \(A\) goes first. Find the probability that the shot will occur when \(A\) has the revolver.
In a box of 2009 socks there are blue and red socks. Can there be some number of blue socks that the probability of pulling out two socks of the same colour at random is equal to 0.5?
Hannah and Emma have three coins. On different sides of one coin there are scissors and paper, on the sides of another coin – a rock and scissors, on the sides of the third – paper and a rock. Scissors defeat paper, paper defeats rock and rock wins against scissors. First, Hannah chooses a coin, then Emma, then they throw their coins and see who wins (if the same image appears on both, then it’s a draw). They do this many times. Is it possible for Emma to choose a coin so that the probability of her winning is higher than that of Hannah?
Gabby and Joe cut rectangles out of checkered paper. Joe is lazy; He throws a die once and cuts out a square whose side is equal to the number of points that are on the upwards facing side of the die. Gabby throws the die twice and cuts out a rectangle with the length and width equal to the numbers which come out from the die. Who has the mathematical expectation of the rectangle of a greater area?
According to the rules of a chess match, the winner is declared to be the one who has beaten their opponent by two defeats. Draws do not count. The probability of winning for both rivals is the same. The number of successful games played in such a match is random. Find its mathematical expectation.