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?
There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.
One of \(n\) prizes is embedded in each chewing gum pack, where each prize has probability \(1/n\) of being found.
How many packets of gum, on average, should I buy to collect the full collection prizes?
Every evening Ross arrives at a random time to the bus stop. Two bus routes stop at this bus stop. One of the routes takes Ross home, and the other takes him to visit his friend Rachel. Ross is waiting for the first bus and depending on which bus arrives, he goes either home or to his friend’s house. After a while, Ross noticed that he is twice as likely to visit Rachel than to be at home. Based on this, Ross concludes that one of the buses runs twice as often as the other. Is he right? Can buses run at the same frequency when the condition of the task is met? (It is assumed that buses do not run randomly, but on a certain schedule).
When boarding a plane, a line of \(n\) passengers was formed, each of whom has a ticket for one of the \(n\) places. The first in the line is a crazy old man. He runs onto the plane and sits down in a random place (perhaps, his own). Then passengers take turns to take their seats, and in the case that their place is already occupied, they sit randomly on one of the vacant seats. What is the probability that the last passenger will take his assigned seat?
Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.
The frog jumps over the vertices of the hexagon \(ABCDEF\), each time moving to one of the neighbouring vertices.
a) How many ways can it get from \(A\) to \(C\) in \(n\) jumps?
b) The same question, but on condition that it cannot jump to \(D\)?
c) Let the frog’s path begin at the vertex \(A\), and at the vertex \(D\) there is a mine. Every second it makes another jump. What is the probability that it will still be alive in \(n\) seconds?
d)* What is the average life expectancy of such frogs?
Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.
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.