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.
James furiously cuts a rectangular sheet of paper with scissors. Every second he cuts a random piece by an unsystematic rectilinear cut into two parts.
a) Find the mathematical expectation of the number of sides of a polygon (made from a piece of paper) that James randomly picks up after an hour of such work.
b) Solve the same problem if at first the piece of paper had the form of an arbitrary polygon.
In the Valley of the Five Lakes there are five identical lakes, some of which are connected by streams (in the image, dotted lines denote the possible “routes” of streams). Small carps are born only in lake \(S\). While the carp is growing up, it passes exactly four times from one lake to another by some stream (the carp chooses a stream at random), and then it remains in the lake in which it ended up. Of every thousand carps, an average of 375 remain in lake \(S\), and the rest remain in lake \(B\), no one else lives in the other lakes. Determine how many streams there are in the Valley of the Five Lakes.
A regular dice is thrown many times. Find the mathematical expectation of the number of rolls made before the moment when the sum of all rolled points reaches 2010 (that is, it became no less than 2010).
The point \(O\) is randomly chosen on piece of square paper. Then the square is folded in such a way that each vertex is overlaid on the point \(O\). The figure shows one of the possible folding schemes. Find the mathematical expectation of the number of sides of the polygon that appears.