You absolutely can manufacture unfair coins either using a two surfaced approach (like a weak magnetic field acting on a magnetized coin) but even a one surfaced approach is possible if you make use of carvings on the surface of the coin to get a favorable result from air resistence - lastly you can even achieve it through density, if one side of a coin is significantly more dense then the other then it will tend to land face down - you can play with this a bit by trying to flip a weighted cylinder and observing the landing pattern - I might suggest taking a coin roll (like you get in a bank) and gluing some coins into one end of it - then try and flip it in the air so it lands coin-side up.
It is, however, very hard to bias a coin significantly without skewing the dimensions or having clear alterations visible on the coin.
https://izbicki.me/blog/how-to-create-an-unfair-coin-and-pro...
Aka, you need to pick a threshold for bias that’s interesting before designing the experiment. A casino might care a great deal about say 0.5% bias which would take a lot of trials to detect.
Interesting results, I thought they'd be more skewed.
Not sure what your point is here. Yes, that is the meaning that is being discussed. You understood it yourself. Grandparent understood it as well. What value do you feel bringing up this point brings to the debate?
> like a weak magnetic field acting on a magnetized coin
Probably not. The field will act through the other face of the coin as well. The coin is pulled towards the magnetic surface, but it doesn't alter the revolution of the coin. If the field is sufficiently weak to not pass through the coin then it wouldn't have any impact on the other side either.
The exception would possibly be mu metal or something else that prevents the magentic field from acting on one face entirely, in combination with a strong magnetic field, and I'm still going to lean towards "probably wouldn't work". It would still intermittently pull the entire coin towards the surface, it's not clear that it would counter the rotation of the coin itself.
> even a one surfaced approach is possible if you make use of carvings on the surface of the coin to get a favorable result from air resistence
No, because air resistance is acting on both sides of the coin at once. The air resistance is a constant A+B, not A,B,A,B.
> lastly you can even achieve it through density
No, this is the entire point of the article. The coin doesn't revolve around its geometric center, it revolves around its center of gravity. By changing the density of one side (balsa wood and lead, as I said) you change the center of gravity but the coin itself has the same rate of revolution.
Considering you were the grandparent, I guess that should be somewhat reassuring.
FWIW, I interpreted your use as rigid in the same manner as the GP (of this post).
As far as I’m concerned a electronic bistable generator that is biased through manipulation is effectively a “weighted coin”.. those exist.. and are counter to your claim unless you restrict “weighted coin” to the completely literal.
God no, GP here means the contextual grandparent of the topic, as it always does. GP means the person I responded to. Why in the world would I respond to myself just because "GP"?
If you haven't read the rules, you need to be giving the most gracious reading of the comment chain here. Pretty obvious what the implication was. Try reading again.
> As far as I’m concerned a electronic bistable generator that is biased through manipulation is effectively a “weighted coin”
Oh cool you're trying to impose a change of topic from the ability to literally bias a coin when a normal person flips it randomly, into telling me to defeat... all attempts of people manipulating all RNGs ever?
Super weird angle for a discussion about flipping coins. Not really interesting.
In fact, most methods of coin toss will be influenced by an unbalanced coin in some way. The paper only demonstrates that if you flip a coin with a certain precisely specified method (and catch it midair) - can you be reasonably assured a weighted coin will be unbiased.
See their own referenced book Jaynes, 1996 pp 1003-1007, which I think gives a much clearer explanation of the possibilities.
Note that the NFL for instance does not catch the coin, so there's at least a real world where a coin could be biased.
The important part of that paper is this:
"Examples of how others have flipped and tossed coins show the students how essential it is to carefully describe the experimental process." not just the one detail about angular momentum and CoG.
http://statweb.stanford.edu/~susan/papers/headswithJ.pdf
PS: Of note they where detecting bias in the range of 1% that’s difficult to detect by hand.
It's not even clear that's a real bias or just random noise, or an anomaly of their testing setup.
Yes but you don't need such a coin. You can use a perfect coin and consider flipping it many times.
[0] Of course I didn't do any statistical analysis or a huge number of trials for this to really tell if I was able to do it but it felt pretty consistent.
(For best results, make the styrofoam thick relatively to the lead, and/or attach extra aerodynamic surfaces :).)
Tangent: there is no such thing. You can weight a die, you cannot weight a coin.
Intuitively this should make sense because even if you made one side of the coin from lead and the other from balsa wood, all you are doing is changing the center of gravity of the coin. The coin spins about its center of gravity, not the geometric center of the coin, so this makes no difference.
https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf