Mathematics

SorryDB: Can AI Really Prove Real-World Math Theorems? Imagine you’re a mathematician knee-deep in a complex proof, but you hit a wall. Instead of giving up, you jot down a placeholder—“sorry, I’ll finish this later”—and move on. Now, picture AI stepping in to fill those gaps automatically. That’s the promise of SorryDB, a groundbreaking benchmark introduced in the paper “SorryDB: Can AI Provers Complete Real-World Lean Theorems?” (arXiv:2603.02668). This isn’t some abstract academic exercise; it’s a practical testbed pulling “sorry” statements from 78 real GitHub projects, challenging AI to prove theorems that actual mathematicians are working on. ...

Table of Contents Introduction Probability Fundamentals Conditional Probability and the Chain Rule Probability Distributions How LLMs Use Probability From Theory to Practice Common Misconceptions Conclusion Resources Introduction If you’ve ever wondered how ChatGPT, Claude, or other large language models generate coherent text that seems almost human-like, the answer lies in mathematics—specifically, probability theory. While the internal mechanics of these models involve complex neural networks and billions of parameters, at their core, they operate on a surprisingly elegant principle: predicting the next word by calculating probabilities. ...