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?
In the cabinet of Anchuria there are 100 ministers. Among them there are honest and dishonest ministers. It is known that out of any ten ministers, at least one minister is dishonest. What is the smallest number of dishonest ministers there could be in the cabinet?
An ant goes out of the origin along a line and makes \(a\) steps of one unit to the right, \(b\) steps of one unit to the left in some order, where \(a > b\). The wandering span of the ant is the difference between the largest and smallest coordinates of the ant for the entire length of its journey.
a) Find the largest possible wandering range.
b) Find the smallest possible range.
c) How many different sequences of motion of the ant are there, where the wandering range is the greatest possible?
A square is divided into triangles (see the figure). How many ways are there to paint exactly one third of the square? Small triangles cannot be painted partially.
We will assume that the birth of a girl and a boy is equally probable. It is known that in some family there are two children.
a) What is the probability that one of them is a boy and one a girl?
b) Additionally, it is known that one of the children is a boy. What is the probability that there is one boy and one girl in the family now?
c) Additionally, it is known that the boy was born on a Monday. What is the probability that there is one boy and one girl in the family now?
Every day, Patrick the dog chews one slipper from the available stock in the house. Strictly with a probability of 0.5 Patrick wants to chew the left slipper, and with a probability of 0.5 – the right one. If the desired slippers are not present, Patrick becomes upset. How many pairs of the same slippers need to be bought, so that with a probability of not less than 0.8 Patrick does not get upset for an entire week (7 days)?
Find the probability that heads will fall an even number of times, in an experiment in which:
a) a symmetrical coin is thrown \(n\) times;
b) a coin is thrown \(n\) times, for which the probability of getting heads in one throw is \(p(0 < p < 1)\).