Related papers: Daniel Litt's Probability Puzzle
We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings,…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
Negative probabilities emerged at intermediate steps in various attempts to predict the distributions of quantum interference. There is no consensus on their meaning yet. It has been suggested (Khrennikov, 1998) that negative probabilities…
The title of the article is identical to the title of Chapter 21 in Gardner (2001): because we are going to analyze the probability calculations and the ambiguity of the problem statements. We will analyze 3 out of 4 problems from Gardner…
Let $G$ be a finite Abelian group of order $d$. We consider an urn in which, initially, there are labeled balls that generate the group $G$. Choosing two balls from the urn with replacement, observe their labels, and perform a group…
Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $\alpha$ balls of the same colour and $\beta$ balls of the opposite colour. Both cases, $\beta=0$ and $\beta>0$ are well…
These lectures introduce key concepts in probability and statistical inference at a level suitable for graduate students in particle physics. Our goal is to paint as vivid a picture as possible of the concepts covered.
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.
A new representation of Dirac's delta-distribution, based on the so-called q-exponentials, has been recently conjectured. We prove here that this conjecture is indeed valid.
We revisit the game in which each of several players chooses a pattern and then a coin is flipped repeatedly until one of these patterns is generated. In particular, we demonstrate how to compute the probability of any one player winning…
We present a variation of the water puzzle, which is related to a simple model of marginal utility. The problem has an intriguing solution and can be extended in several directions.
This paper is motivated by the questions of how to give the concept of probability an adequate real-world meaning, and how to explain a certain type of phenomenon that can be found, for instance, in Ellsberg's paradox. It attempts to answer…
Charles L. Dodgson, also known as Lewis Carroll, in his book "Pillow problems" from 1893 asked for the likelihood of a random triangle to be obtuse. Clearly, the answer to Dodgson's question depends strongly on the assumed random…
We survey the state of the union-closed sets conjecture.
The game of plates and olives was originally formulated by Nicolaescu and encodes the evolution of the topology of the sublevel sets of Morse functions. We consider a random variant of this game. The process starts with an empty table.…
We propose a refined version of the Sato-Tate conjecture about the spacing distribution of the angle determined for each prime number. We also discuss its implications on $L$-function associated with elliptic curves in the relation to…
A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that…
Complex continuous or mixed joint distributions (e.g., P(Y | z_1, z_2, ..., z_N)) generally lack closed-form solutions, often necessitating approximations such as MCMC. This paper proposes Indeterminate Probability Theory (IPT), which makes…
In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random…