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?
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.
Anna is waiting for the bus. Which event is most likely?
\(A =\{\)Anna waits for the bus for at least a minute\(\}\),
\(B = \{\)Anna waits for the bus for at least two minutes\(\}\),
\(C = \{\)Anna waits for the bus for at least five minutes\(\}\).
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.
A cube is created from 27 playing blocks.
a) Find the probability that there are exactly 25 sixes on the surface of the cube.
b) Find the probability that there is at least one 1 on the surface of the cube.
c) Find the mathematical expectation of the number of sixes on the surface of the cube.
d) Find the mathematical expectation of the sum of the numbers that are on the surface of the cube.
e) Find the mathematical expectation of a random variable: “The number of different digits that are on the surface of the cube.”
Peter and 9 other people play such a game: everyone rolls a dice. The player receives a prize if he or she rolled a number that no one else was able to roll.
a) What is the probability that Peter will receive a prize?
b) What is the probability that at least someone will receive a prize?
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?