Fair coins tend to land on the side they started (2023)

Easy way to get a fair result from an unfair coin toss: Flip the coin twice in a row, in this case starting with the same side facing up both times, so it's equally unfair for both tosses. If you get heads-heads or tails-tails, discard and start over until you get either heads-tails or tails-heads, which have equal probabilities (so you can say something like HT = "heads" and TH = "tails").

This works even if the coin lands heads 99% of the time, as long as it's consistent (but you'll probably have to flip a bunch of times in that case).

Importantly - you don't have to know the odds of the coin ahead of time, or which side is more likely. You only need to know that it is consistent.

I’ve seen this attributed to John von Neumann, of all people

What if consecutive unfair coin flips are not independent?

If anyone wants to look up why this might work, it's a Whitening transform [0]. I can't find the name of the algorithm itself being describe in the parent but there's more than just that for accomplishing the same thing.

0: https://en.wikipedia.org/wiki/Whitening_transformation

It seems like he did everything! I first heard of Von Neumann in international relations & economics classes as the person who established game theory, then later in CS classes as the creator of mergesort, cellular automata, Von Neumann architecture, etc.

Then it's impossible to trust the coin in the general case.

Proof: Imagine the extreme case of the coin containing AI that knows exactly how you use it and how to manipulate each toss result. The coin itself can decide the outcome of your procedure, so it's impossible to trust it to generate randomness.

The odds are important to know because if someone gave you a trick coin that always lands on heads, you will be flipping coins until the end of the universe. And I'm sure you have more important things to do than that.

It's also impossible to prove that a given coin is not being controlled by an AI. (Or a deity.)

Wait til you hear about what he did in Math and Physics...

Very easy to claim he was the most intelligent human to ever live. Or perhaps he was never human...

https://en.wikipedia.org/wiki/The_Martians_(scientists)

Nah, you can put in a rule to stop. It would be better to know ahead of time, but you don't need to.

> you will be flipping coins until the end of the universe

Reminds me of one of my favorite movies, Rosencrantz & Guildenstern Are Dead, which opens with just such a scenario[1].

[1]: https://www.youtube.com/watch?v=RjOqaD5tWB0

Yes, which is why you can only trust abstract coins that exist in a formal system which assumes independent tosses :)

If you require true randomness without any assumptions this is not the universe for you.

I consider LLMs to be the first successful non-von-neumann architecture in many decades

Thank you. This was useful to learn.

Just perform the same coin toss in two universes.

Each flip would need to start with the same side up though, if this paper is correct.

That's Von Neumann Whitening.