Problem 1.

There are four nonstandard but fair dice. Each of them has some non-negative integer numbers written on its faces (they could be different for different dice). Suppose we roll them, and we get numbers $A, B, C, D$. Is it possible that \[ \mathbb{P}(A > B) = \mathbb{P}(B > C) = \mathbb{P}(C>D) = \mathbb{P}(D>A) > 1/2 ? \]

Problem 2.

Let $X$ be some non-negative random variable that takes one of the finitely possible many values. Prove that then for any real $a > 0$ it is true that $ \mathbb{P}(X \geq a) \leq \frac{\mathbb{E} X}{a} $.

Problem 3.

Let $X$ be a non-negative random variable that takes one of the finitely possible many values. Prove that $ \mathbb{E}(X^2) \geq \mathbb{E}(X)^2 $.

Problem 4.

There are $n$ personal letters to be sent to $n$ people (one letter for each person). Imagine we do it at random (instead of thinking). What is the mathematical expectation of the number of people who will receive the letters meant for them?