A Return-Probability Perspective on the St. Petersburg Paradox

The classical St. Petersburg paradox is based on the infinite expected value of a simple coin-toss game. Mathematically, this suggests that a rational player should be willing to pay any amount to participate. In practice, however, no one would pay a fortune for such a game. This paper presents a simple alternative approach: instead of relying on expectation, we examine the probability that the payout exceeds the entry fee. This criterion yields an intuitive threshold: once the entry fee surpasses 4, the probability of breaking even drops below 50%. For higher fees, the probability declines rapidly (e.g., only 0.39% for X = 500). This perspective reconciles mathematical theory with everyday intuition and decision-making. It also highlights how different, yet equally correct, mathematical approaches (expected value vs. return-probability) can lead to opposite conclusions.