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Fair coins tend to land on the same side they started

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Re: Fair coins tend to land on the same side they started

#261

Post by extraduder_ire »

jbandela1 wrote: post #16

Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...

I take it a double headed coin doesn't count as biased for this algorithm. (because it's too biased to still be considered a "coin", for probability's sake.)

Otherwise it never halts.

Re: Fair coins tend to land on the same side they started

#262

Post by Terr_ »

vikingerik wrote: post #146
jbandela1 wrote: post #16

Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...

Also interestingly, this extends beyond a two-sided coin, to any number of possible results, like a die with N sides. To get a fair result from a biased dN: Roll it N times. If you don't get all N distinct results, restart. If you do, then the first of those is your final result.

Also-related are all the problems like "simulate an 11-sided die with a 6-sided die", where the solution involves identifying certain rolls that should trigger a retry. [0]

In both cases it has to do with shaping the combination of multiple rolls using some knowledge of outcome-symmetries and overlaps.

____

[0] In this particular case, that's something like: Roll the die twice to get rolls A and B; map those a number X in the range range [1,36] using x=(((A-1)*6)+B); if X33 then start over.

Re: Fair coins tend to land on the same side they started

#263

Post by gala8y »

bryanbuckley wrote: post #259
gala8y wrote: post #6

Earlier quoted context omitted.

This was my intuition in childhood. If you choose tails to be yours and start with tails then catch it, it is most likely to be tails. I came up with this observation myself. Weird.

Noticed as a kid I could flip a quarter with a certain consistency, so I experimented a bit and quickly got to be >90% accurate with an ordinary (controlled) flip. Pretty simple. In fact I just picked up a quarter and practiced (20+ years out of practice) and have some observations: 1) harder than when I was a kid, my fingers are lot bigger + stronger so it's not as precise from the start. A bigger and heavier coin w…

This. When I tried to comment briefly yesterday that the coin falls on the same side, only then did I remember that I actually did that and that it has nothing to do with physics but rather neuroscience (and innocently bent morality - which was also the object of my internal observations). then I remembered that I had actually considered different coin sizes, but I was never as thorough in my attempts to bend the results as you were. oh... the playgrounds of childhoods...

Re: Fair coins tend to land on the same side they started

#265

Post by fbartos »

avipars wrote: post #264

They actually flipped a coin >350,000 times and the paper has 49 authors. Based on the paper https://arxiv.org/pdf/2310.04153.pdf , they recorded video footage of each coin flip I thought they would have run a computer simulation instead.

Yes, I mentioned that in the original paper but it got eddited for some reason.

Re: Fair coins tend to land on the same side they started

#266

Post by quietbritishjim »

rootusrootus wrote: post #214
quietbritishjim wrote: post #162

Earlier quoted context omitted.

That's the usual way of doing it.

Huh. I thought everyone did it my way. Flip it onto the floor, hunt it down, and see which side came up. I'm mostly serious. I know that's not ideal. But I've never been able to master the art of flipping a coin onto the back of my hand, and even catching it mid-air is hit-or-miss for me. The vast majority of time I or anyone I'm with has flipped a coin it's ended up on the floor.

> I've never been able to master the art of flipping a coin onto the back of my hand

That's not the usual way of doing it. Usually, you catch it in the palm of your throwing hand, and only then put it on the back of your other hand, flipping it once more in the process.

Re: Fair coins tend to land on the same side they started

#267

Post by cantrevealname »

tshaddox wrote: post #225
cantrevealname wrote: post #128

Earlier quoted context omitted.

> Why would you test it? I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities don't change by opening a door. Even from mathematicians and statisticians. People were very insistent that the analytical solution was simple and obvious and that switching doors didn't change anything. The only thing that changed some people's minds was a pr…

I'm curious. When those people see the simulation, do they then go back to the analysis and uncover their mistaken reasoning? Or do they just continue to reject the analysis but begrudgingly accept the outcome of the simulation? The analysis of the Monty Hall problem is so very simple I find it very odd to staunchly reject it but then be persuaded by the simulation.

> is so very simple

Years ago, when the Monty Hall problem was not well known, I've seen with my own eyes that it was hard to convince some very smart people of the correct answer. Indeed, after being convinced through simulation or exhaustive enumeration of the decision tree or some other way, they would go back and see what went wrong with their initial analysis.

Re: Fair coins tend to land on the same side they started

#268

Post by fennecfoxy »

098799 wrote: post #84
ChocMontePy wrote: post #69

Earlier quoted context omitted.

If anyone wants to test it, someone wrote a short code that simulates doing that 100,000 times: https://www.techiedelight.com/generate-fair-results-biased-c... The coin is biased to come up TAILS 80% of the time, but using Von Neumann's method in the program I got HEADS 50.035%, TAILS 49.965%.

Why would you test it? Probability of two heads: p*p Probability of two tails: (1-p)*(1-p) Probability of head followed by tails: p*(1-p) Probability of tails followed by heads: (1-p)*p It's not difficult to notice that if you remove the first two, the last two form a 50/50 distribution

Because that's how science is done, theory then practical?

Maybe it's just a meat space thing, but even if the math gives us the answer it still only feels "final" or "true" to me when we've actually tested something out.

Re: Fair coins tend to land on the same side they started

#269

Post by bigstrat2003 »

benchaney wrote: post #255
bigstrat2003 wrote: post #249

Earlier quoted context omitted.

I don't see how a new contestant has less information, though? They know that one of the two doors contains the prize, which is all the previous contestant knows either.

When one door was opened it revealed information about the other two doors.

But it didn't. Before, we knew that one of those two doors could contain either a prize or a goat. After, we know the same exact thing. No information was gained there.

Re: Fair coins tend to land on the same side they started

#270

Post by bigstrat2003 »

cortesoft wrote: post #250
bigstrat2003 wrote: post #243

Earlier quoted context omitted.

No, I don't think that detail makes it any easier. I know that but I still really can't accept the correctness of the Monty Hall strategy (I have to basically just take it on faith and stop trying to understand it). I was trying to put my finger on why, and I think it's this. After Monty eliminates one of the three doors, then the prize is behind one of the two. If someone were to come in this point, with no prior kn…

I have fun trying to explain this problem. Let me see if I can give an explanation that will help you. So lets say you have just picked a door in the beginning. You know you have a 1/3 chance of being right. If I then tell you, "I will give you two options... you can either bet you are right, or bet that you are wrong" You would obviously choose to bet you are wrong, correct? Because you know you only have a 1/3 chan…

That does help - if nothing else, adding up the outcomes manually helps to demonstrate it in a way that's not really easy to grasp on first glance. It still is kind of a spooky result which makes no sense, but at least I feel a little better about the correctness.
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