Math reveals the one game of chance you should always accept

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I challenge you to a game. We roll a fair die, and if it comes up on a 1 or 2, you get $10. If it lands on 3, you get $20. Otherwise, you go home empty-handed. Because you can’t lose anything this way, I ask for a $10 bet for each spin. Do you accept?

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You can decide based on your instinct. But there is also a systematic way to determine whether the risk is worth it. For example, you can consider probability theory. A typical die has six sides, so there are six outcomes. In only two of the six possibilities (i.e. ¹⁄₃ time), you earn $10. Additionally, the chance of winning $20 (i.e. the die lands on 3) is ¹⁄₆. If you multiply these probabilities by the amounts of money and add them up, you get ⁴⁰⁄₆ = ²⁰⁄₃. This means that, statistically speaking, you win on average $6.66 per game.

But I required a bet of $10 per roll. This means I will earn an average of $3.33 per game. All things considered, you should refuse.

Already in 1713, the mathematicians Nicolas I Bernoulli and Pierre de Montmort exchanged ideas on a slightly more complex scenario. It involved flipping a coin until it came up heads for the first time. In this idea, the more throws you make, the more you will win: the winnings are always doubled. So the first time you get tails you get $1, and the next few times you get $2, then $4, and so on. Imagine I offer to play this game with you and ask for an extremely high bet of $2,000. Are you going to accept?

Play at any stake

It is likely that any sane person would answer with a clear “no”. But what would be a reasonable stake? To do this, you can consult the mathematics and look at the expected value: with a probability of ¹⁄₂the coin lands on tails on the first toss, getting tails twice in a row corresponds to a probability of ¹⁄₄doing it three times corresponds to a probability of ¹⁄₈and so on.

At the same time, victory double in each case. The expected value is therefore the infinite sum:

This means that, mathematically speaking, no stake is too high: you still have to play the game.

Because Bernoulli and Montmort chose a St. Petersburg casino as the setting for their thought experiment, they now called this counterintuitive result the St. Petersburg paradox. But this does not denote a paradox in the strict sense; the only paradox is that people would probably never follow the recommended course of action.

These results are counterintuitive in part because they imply infinity. The expected value results from adding an infinite number of sums, so the profit increases quickly. If you succeed in six rolls, you receive $32; if you have six more, your winnings are $2,048. If you get lucky and come up heads six more times, you get $131,072.

The game, in short, is extremely unrealistic: it only works if the challenger has infinite resources. Personally, no. Even casinos with bulging cash registers have an upper limit. Assuming limited capital, the game cannot continue indefinitely.

Set boundaries and engage in a billion-dollar battle

Let’s say I have $1,050 in my account and I’m willing to bet anything to challenge you to the drawing. I can’t ask you to bet $2,000 when the most you’ll get from me is a little over a thousand dollars. So I’m offering you a friendly bet of $6 to play. Do you accept?

Since I only have $1,050, the expected value of the game changes. If you roll 11 numbers in a row, I already owe you $1,024, so I may not be able to finance a 12th roll. Therefore, the changed expected value is now 1 × ¹⁄₂ + 2 × ¹⁄₄ + 4 × ¹⁄₈ + … + 1024 × ¹⁄_2,048 = ¹⁄₂ × 11 = 5.5.

Given my limited wealth, the situation has completely changed. Instead of an infinite expected value, you now get the result $5.5. And that means that at $6 bet, you should refuse the game. On the other hand, if you trade me down to $5 (taking advantage of the fact that I’m not particularly good at mental arithmetic), then your chances of making a profit are good.

What if a billionaire challenged you to play? To calculate the stake you are willing to play for, you must again determine the maximum number of spins this big cat can last before going bankrupt. To get an idea of ​​the scale, after just 38 rounds, your challenger already owes you more than $137 billion. (That number would make Warren Buffett uncomfortable about flipping the coin again and would bankrupt Bill Gates.) Assuming that’s all your opponent can offer, the expected value (the maximum bet you should accept) works out to just $19.

This article was originally published in Spektrum der Wissenschaft and has been reproduced with permission. It was translated from the original German version with the help of artificial intelligence and reviewed by our editors..