A fly moves from the origin only to the right or upwards along the lines of the integer grid (a monotonic wander). In each node of the net, the fly randomly selects the direction of further movement: upwards or to the right.
a) Prove that sooner or later the fly will reach the point with abscissa 2011.
b) Find the mathematical expectation of the ordinate of the fly at the moment when the fly reached the abscissa 2011.
In his laboratory, the Scattered Scientist created a unicellular organism, which, with a probability of 0.6 is divided into two of the same organisms, and with a probability of 0.4 dies without leaving any offspring. Find the probability that after a while the Scattered Scientist will not have any such organisms.
We are given 101 rectangles with integer-length sides that do not exceed 100.
Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).