The First Real Paradox of Probability

In 1713, Nicolaus Bernoulli described a simple coin-flip game. The expected value of this game is infinite, meaning a "rational" player should pay any price to play. But your gut says $10, max. And your gut has a point. This is the story of probability's first great paradox: what it reveals about expected value, rare events, and every decision you make under uncertainty. This puzzle didn't just create a paradox in probability. It led to three centuries of attempts to find a solution that satisfies everyone, and still, none does. From utility theory to ergodicity economics, we'll see how mathematicians have tried to resolve it and why it still matters today. WHAT'S INSIDE 00:00 – Intro: the game that broke probability 02:19 – Chapter 1. The Bet 06:55 – Chapter 2. The 300-Year Hunt for Solutions 16:45 – Chapter 3. Why It Matters 18:54 – Chapter 4. Three Takeaways WHAT ELSE TO READ Daniel Bernoulli's original 1738 paper, in the St. Petersburg Academy journal that gave the paradox its name: https://archive.org/details/commentar... The difficult birth of stochastics (Historia Mathematica): https://www.sciencedirect.com/science... Ole Peters, "The time resolution of the St. Petersburg paradox" (2011), the ergodicity argument: https://arxiv.org/pdf/1011.4404 V.I. Yukalov, "A Resolution of St. Petersburg Paradox" (2021), the stochastic decision-theory approach: https://arxiv.org/abs/2111.14635 Claudio Mattalia, "The Saint Petersburg Paradox and Its Solution" (Risks, 2025): https://www.mdpi.com/2227-9091/13/2/32 Nassim Nicholas Taleb, "The Black Swan" THIS CHANNEL Vital Math explores the beauty of mathematics: paradoxes, probability, and the ideas that quietly run the modern world. Subscribe and turn on notifications so the algorithm's expected value works in your favor. Tell me in the comments: what's YOUR honest price for one round of this game? #mathematics #Probability #Paradox #StPetersburgParadox #ExpectedValue