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

This is probably just because the coins aren’t actually fair. If the coin is slightly biased towards heads, the first throw is more likely to heads, and so are all subsequent throws. Same for tails.

That's not the problem. You can test that by using a highly secure random number generator, e.g. /dev/random in Linux, to select the initial side. Keep track of that initial side, record the side it lands on. This paper shows a same-side bias, not a heads bias.

That's the opposite of what the paper says. If the coin was biased you'd expect it to land on heads more often regardless of what side it starts on. The coins land on the side they start on more often.

No, first of all due to imperfections in the manufacture of real coins, there are actually no fair coins. Also the bias in the probability affects the first throw as well as all the rest. If your dataset is composed of first throws/rest of the throws, you’re going to see they are correlated.

A same side bias is either a heads bias or a tails bias.

Its not, its a bias towards which side the coin started on.

Which is either heads or tails.

How? I described how to randomize the initial side. Boolean true for heads, boolean false for tails, for example. Keep pulling those from the Kernel's secure RNG.

I think you're missing the fact that you don't have to chain coin flips literally right after another.

As the other commenter said, in between coin flips, use a highly secure PRNG to orient the coin randomly. This would correct for your bias (if true).

A coin with a heads bias is more likely to land on heads no matter how it's thrown.

A coin with a same side bias is more likely to land on heads if it's thrown with heads facing up, and more likely to land on tails if thrown with with tails facing up.

If you take a specific coin and find that when you prepare it to be flipped showing heads up, that it is more likely to land heads up, and that when you prepare it to be flipped tails up, it is more likely to land tails up, it seems confusing to call that coin 'heads or tails biased'

You're missing the point.

A coin that is biased towards heads is one that would more often land on heads regardless of how you hold it when you start the flip.

The study finding is that every coin is more likely to land on heads if you start it with heads facing up, and will also be more likely to land on tails, if you start it that way instead. This bias, while small, is greater than the typical observed bias due to imperfections in manufacturing.

It's not about the "first throw" vs the "rest of the throws". It's about how you hold the coin when you go to flip it. That's what they mean by "started".