Problems

Valerie wrote the number 1 on the board, and then several more numbers. As soon as Valerie writes the next number, Mike calculates the median of the already available set of numbers and writes it in his notebook. At some point, in Mike’s notebook, the numbers: 1; 2; 3; 2.5; 3; 2.5; 2; 2; 2; 2.5 are written.

a) What is the fourth number written on the board?

b) What is the eighth number written on the board?

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.”

On a calculator, there are numbers from 0 to 9 and signs of two actions (see the figure). First, the display shows the number 0. You can press any key. The calculator performs the actions in the sequence of clicks. If the action sign is pressed several times, the calculator will only remember the last push. The Scattered Scientist pressed a lot of buttons in a random sequence. Find approximately the probability with which the outcome of the resulting chain of actions is an odd number?

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.

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?