The Chevalier de Mere Problem

A 17th-century French gambler, the Chevalier de Mere, noticed something that bothered him: betting on rolling at least one six in four rolls of a single die was profitable over time, but the seemingly "equivalent" bet on rolling at least one double-six in 24 rolls of two dice was a loser. His intuition said the two bets should have the same odds, since $4$ is to $6$ as $24$ is to $36$ (the same ratio), but experience said otherwise — and he asked Blaise Pascal to explain why, sparking a correspondence with Pierre de Fermat that's often credited as a founding moment of formal probability theory.

The resolution is a straightforward complement calculation. The probability of at least one six in four rolls is $1 - (5/6)^4 \approx 0.518$, a favorable bet. The probability of at least one double-six in 24 rolls of two dice is $1 - (35/36)^{24} \approx 0.491$, an unfavorable bet. The two numbers look close but land on opposite sides of even odds — de Mere's linear-ratio intuition (that scaling both the target and the number of rolls by the same factor preserves the odds) was simply wrong, because the correct calculation involves a probability raised to a power, not a ratio.

The puzzle endures as an interview question because it's a compact test of whether a candidate reaches for the complement rule (compute the probability of the event not happening, then subtract from 1) rather than trying to directly enumerate "at least one" outcomes, and whether they're suspicious enough of a "looks the same" ratio to actually run the numbers.