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Imagine a game where we roll a fair die. If it lands on 1 or 2, you receive $10; if it lands on 3, you collect $20. For any other outcome, you leave with nothing. Since you can’t lose any money, I propose you bet $10 on each roll. Are you in?
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You might rely on intuition to decide, but there’s also a systematic approach to evaluate the risk. Probability theory offers a way forward. A standard die has six sides, offering six possible results. Two of these outcomes (1/3 of the time) earn you $10. The chance of hitting $20 (rolling a 3) is 1/6. Calculating these probabilities against the winnings, the expected average gain is 40/6 = 20/3, which is about $6.66 per game.
However, with a $10 betting stake per roll, you’ll lose an average of $3.33 per game. Therefore, it’s wise to decline the offer.
In 1713, mathematicians Nicolaus I Bernoulli and Pierre de Montmort discussed a more intricate scenario involving a coin toss, continuing until the first head appears. With each toss, the winnings double: starting at $1, then $2, $4, and so forth. Suppose I propose this game to you, asking for a $2,000 stake. Would you agree?
Playing at Any Stake
Most rational people would likely decline. But what would be a sensible stake? Mathematics provides guidance through expected value: a coin landing tails on the first toss carries a 1/2 probability, tails twice in a row equates to 1/4, and three consecutive tails to 1/8, and so on.
The winnings double each time, resulting in an infinite sum for the expected value:
Thus, mathematically, no stake is too great—you should always engage in the game.
This thought experiment, set in a Saint Petersburg casino by Bernoulli and Montmort, is termed the Saint Petersburg paradox. It’s not a true paradox; the real surprise is that people are unlikely to follow the mathematical advice.
These findings are counterintuitive partly due to their infinite nature. The expected value is derived from summing an infinite series, causing rapid profit growth. Six consecutive successes yield $32; six additional successes bring $2,048, and six more result in $131,072.
The game, in essence, is highly unrealistic: it requires infinite resources, which I lack. Even wealthy casinos have financial limits. With finite resources, endless gameplay isn’t feasible.
Setting Limits, and Engaging in a Billion-Dollar Battle
Suppose I have $1,050 available and am eager to challenge you with the coin toss. I cannot request a $2,000 stake when I can offer slightly over $1,000. So, I propose a modest $6 wager. Are you interested?
With only $1,050 at my disposal, the game’s expected value shifts. If you manage 11 consecutive tails, I owe you $1,024, rendering a 12th round financially unfeasible. The revised expected value is 1 × 1/2 + 2 × 1/4 + 4 × 1/8 + … + 1,024 × 1/2,048 = 1/2 × 11 = 5.5.
Given my restricted resources, the scenario transforms completely. Instead of infinite expected value, you now attain $5.5. Therefore, you should decline the $6 stake. However, if you negotiate down to $5, leveraging my weak arithmetic skills, you stand a better chance of profiting.
What if a billionaire invited you to play? To figure out a suitable stake, calculate how many rounds the affluent opponent can last before bankrupting. For perspective, after 38 rounds, they would owe over $137 billion. Even wealthy individuals like Warren Buffett or Bill Gates would hesitate to continue. Assuming that’s the extent of their wealth, the expected value (your maximum stake) is just $19.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
