Problem 1.

There are 3 types of socks: red, green and blue, and there are three boxes: first box is filled with green and red socks, second one is filled with green and blue socks, and the third one is filled with blue and red socks. Each of them has a lot of socks with a 1:1 ratio of colours. You can see the boxes, but you do not which box is which. You are allowed to blindly take a sock from a box of your choice as many times as you like, and your goal is to understand which box has what types of socks (you take socks one by one, you can stop at any time). What would be your strategy to minimise the expected number of times you take a sock?

Problem 2.

Given $n$ points drawn randomly on the circumference of a circle, what is the probability that they are all within a semicircle?

Problem 3.

There is a number 1 written on a board. Every minute a person walks in, and replaces the number on the board, say $x$, with the number $\frac{1}{x}$ or the number $x+1$ (with equal probabilities). Estimate the expected value of the number on the board after 10 minutes.

Problem 4.

There is a square $8 \times 8$ board. Player A is standing in its left-bottom corner, while player B is standing in its upper-right corner. Every second A randomly moves one cell up or one cell to the right, while B randomly moves one cell down one cell to the left. Neither of them can go beyond the borders of the board, so if they reach it, they keep on going in one of the two directions they actually can. When they reach the opposite corner, they stop moving. What is the probability of A and B meeting at some point of time?

Problem 5.

A line of 100 hundred passengers are waiting to board on a plane. For convenience, let's says that $i$-th passenger has a ticket for place number $i$. Being drunk, the first person takes a seat at random. But all other 99 passengers are sober, so they will go to their proper seat unless it is already take. In that case, they will randomly choose a free seat. What is the probability that the person number 100 will get to his/her own seat?

Problem 6.

a) There is a game where one person rolls a 20-sided dice and another person rolls three 6-sided dice. They both roll their dice and whoever gets a sum of numbers wins the game. Which person would you rather be? Is it a fair game?
b) Same game with one more player C who has a 20-sided dice. Is this new game fair?

Problem 7.

a) Paula shuffles a deck of cards thoroughly, then plays cards face up one at a time, from the top of the deck. At any time Victor can interrupt Paula and bet £1 that the next card will be red (if he never interrupts, he's automatically betting on the last card). What's Victor's best strategy? How much better than 50%-50% of winning can he do? Assume there are 26 red and 26 black cards in the deck.
b) Again Paula shuffles a deck thoroughly and plays cards face up one at a time. Victor begins with a bankroll of £1, and can bet any fraction of his current worth, prior to each revelation, on the color of the next card. He gets even odds regardless of the current composition of the deck. Thus, for example, he can decline to bet until the last card, whose colour he of course knows, then bet everything and be assured of going home with £2.
Is there any way Victor can guarantee to finish with more than £2? If so, what’s the maximum amount he can assure himself of winning?

Problem 8.

You are given a black-box which returns a random number between 0 and 1 (with uniform distribution). You keep generating random numbers, and store the sum of all those random numbers. You stop as soon as the sum exceeds 1. What is the expected number of numbers used in the process?