probability

There are a number of approaches to apply category theory to probability and related fields, such as statistics, information theory and dynamical systems. On one hand, one can study the existing structures in traditional probability theory (such as probability spaces, integration, and so on) using a categorical lens. For instance, the Giry monad models the formation of spaces of probability measu…

A stochastic process describes a dynamical system evolving over a linearly ordered set (“time”), typically taken to be the (positive) integers or real numbers, whose dynamical laws of motion are morphisms in the Kleisli category of the Giry monad (or any other probability monad). By working in the larger category of algebras of that monad, a characterization of a stochastic processes can be model…

Two players start 1 meter away from a target. They simultaneously begin moving towards the target at a same constant speed. If the left player shoots when he is X meters from the target, his shot hit with a probability 1-X... Read more

Question: Let $K$ be a compact Hausdorff space. Is there always a strictly positive Borel probability measure? Thoughts: If $K$ is metrizable it contains a dense countable subset, so the sum over all ...

How large would the following number be? The minimum number of digits of $b^{1/b}$ in base $b$ that are necessary for each number from $0$ to $b$ to occur the same number of times. For example in base ...

Nature, Published online: 01 June 2026; doi:10.1038/d41586-026-00821-4 An experiment with 2,520 participants backs Richard Feynman’s answer to every diner’s dilemma: do I want to try something new?

Physicist Richard Feynman turned a lunch dilemma into a math problem. Researchers finally cracked his notes and found people approximate his solution on their own.

Horsten and Brickhill (2024) propose two methods for constructing non-Archimedean probability functions on the set-theoretic universe V: the finite snapshot approach and the bootstrapping approach. The mathematical execution is competent and the results within their framework are correct. This note raises three foundational concerns that the paper does not adequately address. First, the proposed …

In This Article The Question The Intuition Trap: The Base Rate Fallacy The Mathematical Proof Python Simulation: 1,000,000 Trials Litigation Application: When Juries Get the Math Wrong The Question A cab was involved in a hit-and-run accident at night. Two cab companies operate in the city: the Green company and the Blue company. You are given the following facts: 85% of the cabs in the city are …

In This Article The Question The Intuition Trap: Why 50/50 Feels Obvious The Exhaustive Case Proof The Bayesian Derivation The Generalized N-Door Problem Python Simulation: 1,000,000 Trials Business Application: Bayesian Updating Under New Evidence The Question You are a contestant on a game show. In front of you stand three closed doors. Behind one of them is a car; behind the other two are goat…

How Knives Out teaches Bayesian thinking (without you realizing it) The post Solving a Murder Mystery Using Bayesian Inference appeared first on Towards Data Science .

The function (1 + cos(x))/2 gives a fair approximation to the Gaussian density exp(−x²) You can make the approximation much better by raising it to a power. The function ((1 + cos(x))/2)4 gives a good lower bound and ((1 + cos(x))/2)3.5597 gives a good upper bound. More on that here. There are other ways of […] The post Another Gaussian approximation first appeared on John D. Cook .

_Sitting with a Mad Mind : A Working Thesis_. forthcomingCaptured Resonance as Structural Law advances three falsifiable cross-scale predictions. The first — that the kinetics of attractor reconstitution following substrate-directed intervention, normalized for substrate-regeneration capacity, show comparable structural signatures across scales — is the prediction on which the cross-scale identit…

I was trying to explain the Monty Hall problem to a sceptical friend. But this led to an issue that has left me a little confused. I assume that I should describe the Monty Hall problem here, and not ...

I recently encountered the following problem: You roll a six-sided die and keep track of the sum as you roll, until the sum exceeds 100. What is the probability that the second to last roll is a $2$? ...

Statistical modeling of claim severity distributions is central to actuarial science and risk management, where parameter estimation must balance efficiency and robustness. Maximum likelihood estimation (MLE) is asymptotically efficient under correct model specification but sensitive to extreme observations and perturbations from the assumed distributional form. Robust L-estimators, including the…

A cardinal $\kappa$ is real-valued measurable (rvm) if there is a $\kappa$-additive probabilistic measure on $\kappa$ which vanishes on singletons and is moreover atomless, i.e., any positive set can ...

Kullback-Leibler divergence Kullback-Leibler divergence is defined for two random variables X and Y by K-L divergence is non-negative, and it’s zero if and only if X and Y have the same distribution. But it is not a metric, for reasons explained here. For one thing, it’s not symmetric. Jeffreys divergence We can fix the symmetry problem by […] The post Turning K-L divergence into a metric first a…

Scientific Reports, Published online: 27 May 2026; doi:10.1038/s41598-026-55101-y Expected KL risk quantifies when first-order power-law approximations are sufficient

I expected the 2026 World Cup bracket to be a sorting problem. It turned out to be a sorting problem plus a lookup table. I recently worked on a small World Cup 2026 bracket project, and the strangest part was not building the knockout bracket itself. It was handling the third-placed teams. The new format has: 48 teams 12 groups 24 teams qualifying as group winners and runners-up 8 more teams qua…