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
Is it possible to:
a) load two coins so that the probability of “heads” and “tails” were different, and the probability of getting any of the combinations “tails, tails,” “heads, tails”, “heads, heads” be the same?
b) load two dice so that the probability of getting any amount from 2 to 12 would be the same?
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?
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?
Anna, Boris and Fred decided to go to a children’s Christmas party. They agreed to meet at the bus stop, but they do not know who will come to what time. Each of them can come at a random time from 15:00 to 16:00. Fred is the most patient of them all: if he comes and finds that neither Anna nor Boris are at the bus stop, then Fred will wait for one of them for 15 minutes, and if he waits for more than 15 minutes and no one arrives he will go to the Christmas party by himself. Boris is less patient: he will only wait for 10 minutes. Anna is very impatient: she will not wait at all. However, if Boris and Fred meet, they will wait for Anna until 16:00. What is the probability that all of them will go to the Christmas party?