A Probability Puzzle

TL;DR

The paper addresses Litt's urn puzzle, a probability problem about an urn with an unknown number of red balls. It adopts conditional probability together with the law of total probability, modeling the red-ball count $R$ as uniform on {0,...,100} and computing $P(r_2|r_1)$ through $P(r_1\cap r_2)$ and $P(r_1)$. The calculation yields $P(r_2|r_1)=\frac{2}{3}$, demonstrating that observing a red first draw increases the likelihood of a red second draw. This result connects to classical paradoxes such as the Bertrand's box paradox and the Monty Hall problem, and the article notes alternative Bayesian-network perspectives while highlighting the educational value of undergraduate probability techniques for solving brainteasers.

Abstract

In this short article, we present a solution to one of the probability puzzles that Daniel Litt, a mathematician at the University of Toronto, posted on his X account earlier this year. The main goal of this note is to show how some of the typical concepts taught in an undergraduate probability course can be used to solve these types of probability problems, which sound simple, but can be very difficult to solve.