Related papers: A probabilistic way to discover the rainbow
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i. The aim of the paper is twofold - to put order in the multitude of conjectures that relate to…
Lotteries are commonly employed in school choice to fairly resolve priority ties; however, current practices typically keep students uninformed about their lottery outcomes at the time of preference submission. This paper advocates for…
Two photons are said to be identical when they are prepared in the same quantum state. Given the latter, there is a unique way to achieve this. Conversely, there are many different manners to prepare two non-identical photons: they may have…
This article, based on a talk, treats some elementary, but not completely simple examples from probability. They concern multiple birthday coincidences, throwing dice, the combinatorics of the German card game "Doppelkopf", and the…
Computing the similarity between two probability distributions is a recurring theme across control. We introduce a unified family of distances between the probability distributions of two random variables that is based on the discrepancy…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
If a line cuts randomly two sides of a triangle, the length of the segment determined by the points of intersection is also random. The object of this study, applied to a particular case, is to calculate the probability that the length of…
We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of…
When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system in which P much greater than L, our natural intuition tells us that the box with the average number of balls has the…
The aim of this paper is to show that the concept of probability is best understood by dividing this concept into two different types of probability, namely physical probability and analogical probability. Loosely speaking, a physical…
We consider training probabilistic classifiers in the case of a large number of classes. The number of classes is assumed too large to perform exact normalisation over all classes. To account for this we consider a simple approach that…
In the present popular science paper we determine when a square can be dissected into rectangles similar to a given rectangle. The approach to the question is based on a physical interpretation using electrical networks. Only secondary…
In the Colored Bin Packing problem a set of items with varying weights and colors must be packed into bins of uniform weight limit such that no two items of the same color may be packed adjacently within a bin. We solve this problem for the…
Interpreting experimental data in high school experiments can be a difficult task for students, especially when there is large variation in the data. At the same time, calculating the standard deviation poses a challenge for students. In…
We present a technique for estimating the similarity between objects such as movies or foods whose proper representation depends on human perception. Our technique combines a modest number of human similarity assessments to infer a pairwise…
The two envelopes paradox is discussed. By calculating the conditional probability, we arrive at a conditional expectations which differs from existing results.
Children, sitting in a circle, each have a nonnegative number of candies in front of them. A whistle is blown and each child with more than one candy passes one candy to the left and one to the right. The sharing process is repeated until a…
Various strategies for active learning have been proposed in the machine learning literature. In uncertainty sampling, which is among the most popular approaches, the active learner sequentially queries the label of those instances for…
In many applications of natural language processing it is necessary to determine the likelihood of a given word combination. For example, a speech recognizer may need to determine which of the two word combinations ``eat a peach'' and ``eat…