all articles
Connecting Distributions
This short note builds intuition for how several core distributions—Poisson, Exponential, Gamma, Beta, Binomial, Chi Square, Dirichlet, Multinomial—are tightly connected. Rather than treating them as isolated formulas, we view them as different lenses on the same random process: events happening in time or within the same family.
Bertrand's Paradox
What’s the probability that a random chord in a circle is longer than the side of an inscribed triangle? Surprisingly, the answer is 1/4… or 1/3… or 1/2. All are correct. Bertrand’s Paradox reveals why “random” is not a mathematical concept until you define the underlying probability measure — and why hidden modeling assumptions shape every simulation you run.
Modified Coupon Collector's Problem
This article explores the coupon collector's problem, analyzing the expected number of trials and probabilities to collect all or a fraction of distinct items, with extensions to modified scenarios and connections to combinatorics, simulations, and the philosophical Ship of Theseus
Birthdays
This article explores the probability of having at least one birthday every day of the year in a group of people, leveraging combinatorics, probability, and simulations to address the problem through various mathematical frameworks, including the Coupon Collector's Problem and Stirling numbers.
Mismatched Socks
This article explores the probability of selecting unmatched socks from a pile using mathematical approaches, simulations, and comparisons of sequential and simultaneous selection methods