Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?
In a box, there are 10 white and 15 black balls. Four balls are removed from the box. What is the probability that all of the removed balls will be white?
Carry out the following experiment 10 times: first, toss a coin 10 times in a row and record the number of heads, then toss the coin 9 times in a row and again, record the number of heads. We call the experiment successful, if, in the first case, the number of heads is greater than in the second case. After conducting a series of 10 such experiments, record the number of successful and unsuccessful experiments. Collect the statistics in the form of a table.
a) Anton throws a coin 3 times, and Tina throws it two times. What is the probability that Anton gets more heads than Tina?
b) Anton throws a coin \(n + 1\) times, and Tanya throws it \(n\) times. What is the probability that Anton gets more heads than Tina?
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.
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 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?