this post was submitted on 12 Oct 2023
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As far as I can tell this is related to Benford's law. The coin has to land sometime and at any given time it has spent as much or more time facing the way up it started, so it has more opportunities to land that way up. The fewer times it spins, the more pronounced the effect.
That accords with the finding that each flipper had a different average bias (if each flipper has a characteristic spin rate).
I remember trying to show this in math class in jr. high. I would deliberately try to flip the coin in the same fashion each time. I was able to get something like a 70/30 split out of 100 flips.
I think that's only relevant for coin flips which last for less than half a spin. Probably much less, considering momentum.
So maybe this means that 0.8% of all coin flips make less than half a flip. They basically just throw the coin up, without changing it's rotation?
I don't see how the number of times it spins could have any effect, as 1 spin is equal to 3 spins, or 101 and 57895.
But it matters if the coin spins at all.
If you understand why it matters that it spins at all, you understand why it matters that on each 360 degree spin, it will be facing up for the first half of the spin. Think it through.
I honestly tried, thanks for the impulse.
Yes, and that's decisive if the coin is stopped during that phase. It will be facing down for the second half of the spin, which is decisive if the coin is stopped in that phase instead.
Since coins can spin with different speeds and can be stopped after different periods of time, this should be somewhat random, once it's spinning.
I think I got your point that on average, the coin is facing more up than down, since it started facing up. But I think that's only relevant if the coin is stopped in that initial phase, before making at least half a spin.
Wait, are we approaching the same argument from different sides? If we assume a random distribution of spinning speeds and a random distribution of toss durations. Then there will be some coins which experience very slow rotation, and which are also caught early enough that they don't complete even half a spin. These have to face up.
All the other combinations of spin and toss produce random faces.
It's not like rolling a die, the toss has to end at a point in time and time is linear. T=1 must happen before T=2. At any given time, the difference between the amount of time spent facing up and facing down will be between 0 and T/2n where T is the total time spent in the air and n is the number of spins completed in T. The more times it spins, the smaller the maximum difference between the two but there will always be a difference. It has more chances to land face up than it does face down.
Get your head around Benford's Law. It's a headfuck but it's true for certain data generating processes. A coin toss doesn't produce the same kind of data but it is the equivalent process for a binary outcome.
E2A: actually, it is a bit like rolling a die (because they come to rest at time T too). But that's a much more complicated problem because the die doesn't just land, it bounces around a bit. There's some stuff out there about this for craps and I guess it's why the dice have to hit the back wall. Any edge due to technique must be small enough for casinos not to care about it.