Ben is going to bend a square sheet of paper \(ABCD\). Ben calls the fold beautiful, if the side \(AB\) crosses the side \(CD\) and the four resulting rectangular triangles are equal. Before that, Jack selects a random point on the sheet \(F\). Find the probability that Ben will be able to make a beautiful fold through the point \(F\).
40% of adherents of some political party are women. 70% of the adherents of this party are townspeople. At the same time, 60% of the townspeople who support the party are men. Are the events “the adherent of the party is a townsperson” and “the adherent of party is a woman” independent?
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?
A fly crawls along a grid from the origin. The fly moves only along the lines of the integer grid to the right or upwards (monotonic wandering). In each node of the net, the fly randomly selects the direction of further movement: upwards or to the right. Find the probability that at some point:
a) the fly will be at the point \((8, 10)\);
b) the fly will be at the point \((8, 10)\), along the line passing along the segment connecting the points \((5, 6)\) and \((6, 6)\);
c) the fly will be at the point \((8, 10)\), passing inside a circle of radius 3 with center at point \((4, 5)\).
In a numerical set of \(n\) numbers, one of the numbers is 0 and another is 1.
a) What is the smallest possible variance of such a set of numbers?
b) What should be the set of numbers for this?
The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions that are thought up and sent in by the viewers of the programme. Envelopes with the questions are selected in turn in random order with the help of a spinning top with an arrow. If the experts answer correctly, they earn a point, and if they answer incorrectly, the viewers get one point. The game ends as soon as one of the teams scores 6 points. The probability of the team of experts winning in one round is 0.6 and there can be no draws. Currently, the experts are losing 3 to 4. Find the probability that the experts will still win.
In the set \(-5\), \(-4\), \(-3\), \(-2\), \(-1\), \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), replace one number with two other integers so that the set variance and its mean remain unchanged.
Alice has six magic pies in her pocket: two magnifying pies (if you eat it, you will grow), and two reducing pies (if you eat it, you will shrink). When Alice met Mary Ann, she, without looking, took out three pies from her pocket and gave them to Mary Ann. Find the probability that one of the girls does not have any magnifying pies.
Prince Charming, and another 49 men and 50 women are randomly seated around a round table. Let’s call a man satisfied, if a woman is sitting next to him. Find:
a) the probability that Prince Charming is satisfied;
b) the mathematical expectation of the number of satisfied men.
In Anchuria, there is a single state examination. The probability of guessing the correct answer to each exam question is 0.25. In 2011, in order to obtain a certificate, it was necessary to answer correctly to 3 questions out of 20. In 2012, the Anchuria School of Management decided that 3 questions were not enough. Now you need to correctly answer 6 questions out of 40. It is asked, if you do not know anything but just guess the answers, in what year is the probability of obtaining an Anchurian certificate higher: in 2011 or 2012?