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.
Louis performs in the USE test in mathematics. The exam consists of three types of assignments: \(A\), \(B\), and \(C\). For each of the tasks of type \(A\), four choices are given, only one of which is correct. There are 10 of such tasks. Tasks of type \(B\) and \(C\) require a written
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?
Peter proposes to Sam the opportunity to play the following game. Peter gives Sam two boxes of sweets. In each of the two boxes are chocolate sweets and caramels. In all, there are 25 candies in both boxes. Peter proposes that Sam takes a candy from each box. If both sweets turn out to be chocolate, then Sam wins. Otherwise, Peter wins. The probability that Sam will get two caramels is 0.54. Who has a greater chance of winning?
On each of four cards there is written a natural number. Take two cards at random and add the numbers on them. With equal probability, this amount can be less than 9, equal to 9 or more 9. What numbers can be written on the cards?
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?
There are 25 children in a class. At random, two are chosen. The probability that both children will be boys is \(3/25\). How many girls are in the class?
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?
Three tired cowboys went into a bar, and hung their hats on the buffalo horn at the entrance. When the cowboys left at night, they were unable to distinguish one hat from another and therefore took the three hats at random. Find the likelihood that none of them took their own hat.
A die is thrown over and over again. Let \(P_n\) denote the probability that, at some point, the sum of the points, taken from the numbers that came out on the top face of the die, from all the rolls made, is \(n\). Prove that for \(n \geq 7\) the equality \(P_n = 1/6 (P_{n-1} + P_{n-2} + \dots + P_{n-6})\) is true.