Problems

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?

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 program. Envelopes with the questions are selected in turn in a 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 scored 6 points. Suppose that the abilities of the teams of experts and viewers are equal.

a) Find the mathematical expectation of the number of points scored by the team of experts in 100 games.

b) Find the probability that, in the next game, envelope number 5 will come up.

On board the airplane there are $2n$ passengers, and the airline loaded for them $n$ servings of lunch with chicken and $n$ servings with fish. It is known that a passenger with a probability of 0.5 prefers chicken and with a probability of 0.5 prefers fish. Let’s call a passenger dissatisfied if he does not have what he prefers.

a) Find the most likely number of dissatisfied passengers.

b) Find the mathematical expectation of the number of dissatisfied passengers.

c) Find the variance of the number of dissatisfied passengers.

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?

James bought $n$ pairs of identical socks. For $n$ days James did not have any problems: every morning he took a new pair of socks out of the closet and wore it all day. After $n$ days, James’ father washed all of the socks in the washing machine and put them into pairs in any way possible as, we repeat, all of the socks are the same. Let’s call a pair of socks successful, if both socks in this pair were worn by James on the same day.

a) Find the probability that all of the resulting pairs are successful.

b) Prove that the expectation of the number of successful pairs is greater than 0.5.

On a laundry drying line $n$ socks hang in a random order (the order in which they got out of the washing machine). Among them there are the two favourite socks of the Scattered Scientist. The socks are covered by a drying sheet, so the Scientist does not see them, and takes out one sock by touch. Find the mathematical expectation of the number of socks taken out by the Scientist by the time he has both of his favourite socks.

$N$ pairs of socks hang on a washing line in a random order (the order in which they were taken out of the washing machine). There are no two identical pairs. The socks hang under the drying sheet, so the Scattered Scientist takes out one toe by touch and compares each new sock with all of the previous ones. Find the mathematical expectation of the number of socks taken at the moment when the Scientist will have some pair.

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.