Problems

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%.

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?

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?

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?

On a roulette, any number from 0 to 2007 can be determined with the same probability. The roulette is spun time after time. Let $P_k$ denote the probability that at some point the sum of the numbers that are determined by a ball being thrown on the roulette is $k$. Which number is larger: $P_{2007}$ or $P_{2008}$?

The figure shows the scheme of a go-karting route. The start and finish are at point $A$, and the driver can go along the route as many times as he wants by going to point $A$ and then back onto the circle.

It takes Fred one minute to get from $A$ to $B$ or from $B$ to $A$. It also takes one minute for Fred to go around the ring and he can travel along the ring in an anti-clockwise direction (the arrows in the image indicate the possible direction of movement). Fred does not turn back halfway along the route nor does not stop. He is allowed to be on the track for 10 minutes. Find the number of possible different routes (i.e. sequences of possible routes).

On the occasion of the beginning of the winter holidays all of the boys from class 8B went to the shooting range. It is known that there are $n$ boys in 8B. There are $n$ targets at the shooting range which the class attended. Each of the boys randomly chooses a target, while some of the boys could choose the same target. After this, all of the boys simultaneously attempt to shoot their target. It is known that each of the boys hits their target. The target is considered to be affected if at least one boy has hit it.

a) Find the average number of affected targets.

b) Can the average number of affected targets be less than $n/2$?

An exam is made up of three trigonometry problems, two algebra problems and five geometry problems. Martin is able to solves trigonometry problems with probability $p_1=0.2$, geometry problems with probability $p_2=0.4$, and algebra problems with probability $p_3=0.5$. To get a $B$, Martin needs to solve at least five problems, where the grades are as follows $(A+,A,B,C,D)$.

a) With what probability does Martin solve at least five problems?

Martin decided to work hard on the problems of any one section. Over a week, he can increase the probability of solving the problems of this section by 0.2.

b) What section should Martin complete, so that the probability of solving at least five problems becomes the greatest?

c) Which section should Martin deal with, so that the mathematical expectation of the number of solved problems becomes the greatest?

$N$ people lined up behind each other. The taller people obstruct the shorter ones, and they cannot be seen.

What is the mathematical expectation of the number of people that can be seen?