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 \(n/2\)?
Kate and Gina agreed to meet at the underground in the first hour of the afternoon. Kate comes to the meeting place between noon and one o’clock in the afternoon, waits for 10 minutes and then leaves. Gina does the same.
a) What is the probability that they will meet?
b) How will the probability of a meeting change if Gina decides to come earlier than half past twelve, and Kate still decides to come between noon and one o’clock?
c) How will the probability of a meeting change if Gina decides to come at an arbitrary time between 12:00 and 12:50, and Kate still comes between 12:00 and 13:00?
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?
The building has \(n\) floors and two staircases running from the first to the last floor. On each staircase between each two floors on the intermediate staircase there is a door separating the floors (it is possible to pass from the stairs to the floor, even if the door is locked). The porter decided that too many open doors is bad, and locked up exactly half of the doors, choosing the doors at random. What is the probability that you can climb from the first floor to the last, passing only through open doors?
In the centre of a rectangular billiard table that is 3 m long and 1 m wide, there is a billiard ball. It is hit by a cue in a random direction. After the impact the ball stops passing exactly 2 m. Find the expected number of reflections from the sides of the table.
The mathematics teacher suggested changing the voting scheme at the performance competition. Currently, two groups compete in the final. In the first group, there are \(n\) pupils from class 5A, and in the second, there are \(n\) pupils from class 5B. \(2n\) mothers of all \(2n\) students attended the final of the competition. The best performance is chosen by the mothers voting. It is known that exactly half of the mothers vote honestly for the best performance, and the other half, in any case, vote for the performance in which her child participates (see problem number 65299). The teacher believes that it is necessary to choose a jury of \(2m\) people \((2m \ensuremath{\leq} n)\) from all \(2n\) mums at random. Find the probability that the best performance will win under such voting conditions.
A numerical set \(x_1, \dots , x_n\) is given. Consider the function \(d(t) = \frac{min_{i=1,\dots ,n}|x_i-t| + max_{i=1,\dots ,n}|x_i - t|}{2}\).
a) Is it true that the function \(d (t)\) takes the smallest value at a single point, for any set of numbers \(x_1, \dots , x_n\)?
b) Compare the values of \(d (c)\) and \(d (m)\) where \(c = \frac{min_{i=1,\dots ,n}x_i + max_{i=1,\dots ,n}x_i}{2}\) and \(m\) is the median of the specified set.