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We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here $\text{in …

Every day, I randomly pick a sample consisting of $k$ members of $\{1,\ldots,n\}$ where $k\leq n$. I stop as soon as every number of $\{1,\ldots,n\}$ has been picked at least once. Let $S$ be the numb …

Let $X\neq\emptyset$ be a set and let $\mu:{\cal P}(X)\to [0,1]$ be a probability measure. Is there a probability measure $$\bar{\mu}:{\cal P}({\cal P}(X))\to [0,1]$$ with the following property? …

Let $n\geq 2$ be an integer. We pick $n$ points in $[0,1]$ with uniform distribution. Let $A$ be the minimum distance that two adjacent points have, and let $B$ be the maximum distance that two adjace …

Motivation. In a coin game, a player flips all their coins every turn, starting with just one coin. If the coins all land heads then the game stops; otherwise, the number of coins is doubled for the f …

"Real-life" motivation. The German satirical magazine Der Postillon suggested a few measures for deterring smokers from their bad habit. I especially liked the idea of inserting one "prank cigarette" …

Let $\text{Mat}(\mathbb{N},\{0,1\})$ be the set of all maps $A:\mathbb{N}\times\mathbb{N}\to \{0,1\}$. We define a matrix multiplication for $A, B\in \text{Mat}(\mathbb{N},\{0,1\}$) and $m,n\in\mathbb …

I am given $n$ objects and for $n$ times, I pick one of them with uniform probability and put it back after picking it. For $k\in\{1,\ldots,n\}$ let $f_k$ denote the number of times that I have picke …

I am given $n$ balls. For $n$ times, I pick one of them with uniform probability and put it back after picking it. Let $U$ be the number of balls I have never picked, so $U\in \{0,\ldots,n-1\}$. We a …

Motivation. With my younger son I played the following game on a big (dysfunctional) clock which can be modelled as $\mathbb{Z}/12\mathbb{Z}$ : Put the clock hands at number $12 ( = 0)$. Toss a coin, …

We pick $n\ge 2$ points in $[0,1]$ with uniform distribution. What is the expected value of the largest distance of $2$ adjacent points?

Fix a positive integer $n$. Every second, a particle is sent along a straight line from a fixed position in a fixed direction, at a random integer speed chosen uniformly in $\{1,\ldots, n\}$ meters pe …

Let $E^c_n$ be the expected number of connected components of a simple undirected graph on the vertex set $\{1,\ldots,n\}$. (Every possible edge in $\big\{\{a, b\}: a, b\in \{1,\ldots,n\} \land a \neq …

Here is an example. Let $\mu:{\cal P}(\mathbb{N})\to[0,1]$ be defined by $$\mu(A) := \lim\inf_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} \text{ , for any }A\subseteq \mathbb{N}.$$

I have a die that produces uniformly distributed values in $\{1,\ldots, k\}$ for some integer $k\geq 2$. Now I play the following game. I start rolling the die and produce one integer in $\{1,\ldots, …