A die is thrown six times. Find the mathematical expectation of the number of different faces the die lands on.
A sailor can only serve on a submarine if their height does not exceed 168 cm. There are four teams \(A\), \(B\), \(C\) and \(D\). All sailors in these teams want to serve on a submarine and have been rigorously selected. There remains the last selection round – for height.
In team \(A\), the average height of sailors is 166 cm.
In team \(B\), the median height of the sailors is 167 cm.
In team \(C\), the tallest sailor has a height of 169 cm.
In team \(D\), the mode of the height of the sailors is 167 cm.
In which team, can at least half of the sailors definitely serve on the submarine?
The length of the hypotenuse of a right-angled triangle is 3.
a) The Scattered Scientist calculated the dispersion of the lengths of the sides of this triangle and found that it equals 2. Was he wrong in the calculations?
b) What is the smallest standard deviation of the sides that a rectangular triangle can have? What are the lengths of its sides, adjacent to the right angle?
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.
The upper side of a piece of square paper is white, and the lower one is red. In the square, a point F is randomly chosen. Then the square is bent so that one randomly selected vertex overlaps the point F. Find the mathematical expectation of the number of sides of the red polygon that appears.
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.
A fair dice is thrown many times. It is known that at some point the total amount of points became equal to exactly 2010.
Find the mathematical expectation of the number of throws made to this point.
The Scattered Scientist constructed a device consisting of a sensor and a transmitter. The average life expectancy of the sensor part is 3 years, the average lifetime of the transmitter is 5 years. Knowing the distribution of the lifetime of the sensor and the transmitter, the Scattered Scientist calculated that the average lifetime of the entire device is 3 years 8 months. Was the Scattered Scientist wrong in his calculations?