If you flip a coin enough times...

[NanoStuff]

Theoretical scenario with no statistical noise where a balanced coin would have, to a diminished degree of bias, a 50/50 chance of landing on a particular side and made so that it cannot land in limbo.

The coin however has a certain imprint, say, the shape of a beaver lightly embedded on one of it's sides, giving it a very slight bias to one side. The material is unchanged, the object is simply carved in. The coin is flipped a ridiculous number of times and the probability is recorded as an extremely long floating point value. 50.000002982482...... so on.

Assuming a powerful enough computer(absurdly powerful mind you) that simulates flips of a countless number of differently shaped coins:

After being fed in a an extremely precise probability number, can the computer determine, to a high degree of precision, that the coin with this particular probability has a beaver engraved on one of it's sides?

 

[PolymerTim]

I think that depends on the coins you select and the information you give the computer. For instance, if every coin had a slightly different bias and you gave the computer accurate enough probability (coin flips), then of course it could calculate a bias and match it up with the list of coins you provided (each one with an assigned probability). If you were feeding it coin flips one at a time, the statistical analysis would even tell you when you have enough coin flips to determine that the result distinctly belongs to a particular coin in the list with a given confidence level.

Actually, I saw on TV once some people who put this very concept into action. The case concerns roulette wheels and the fact that, as they are used, the bearing they spin on wears down slightly unevenly due to the uneven force with which they are spun. For this reason, older roulette wheels have a slightly increased chance of giving certain winning numbers. Some guy with a little statistics knowledge decided to spend several weeks in a casino writing down all the winning numbers of the roulette tables and then wrote a program in qbasic to analyze the data.

Sure enough, he found a correlation between certain wheels and their winning numbers. Then he got his whole family in on the record keeping and they collected tens of thousands of winning numbers over the next few weeks until he had enough data to meet a desired confidence level. Then, along with his entire extended family, they started playing the wheels with the known winning numbers. They won some and lost some, but the technique assured that the odds were slightly stacked in their favor and with a big enough investment, they were starting to see significant winnings.

Here's my favorite part. The casino figured out what they were doing and decided to rearrange the roulette wheels on the tables. After one night of losses, it was obvious to the guy what had happened and he went back to recording mode. Recording only one more night of winning numbers he was able to quickly determine which roulette wheels had been moved to which tables and he was back in business.

In the end, the casino refused to allow them entry. It ended up in court and the judgment was that the guy had done nothing illegal, but that the casino was private property and they had a right to not let him play if they didn't want him to.

So, its a long story, but I think it is very similar to what you are getting at with the original question. I hope it helps.

-Tim

 

[BrownTown]

Yes, statistically speaking you could define a confidence integral and with a large enough number of flips you could ensure a very high probability of correctly identifying the coin (never 100%, but approaching that asymtotically as the number get REALLY large).

As for the Roulette wheels, I have heard that same scenario as well, but its very difficult (much more so than counting cards) and the probabilities are only slightly skewed in your favor, so you have to play ALOT of times to win big, and eventually someone is gonna catch on that you keep betting the exact same number on each wheel everytime.

 

[spikespiegal]

As for the Roulette wheels, I have heard that same scenario as well, but its very difficult (much more so than counting cards)

Both the rulette wheel and tossed coin would have to be put in motion with absurd mechanical consistency for countless repetitions for the math to have a chance of working out and producing data for a virtual model. The tossed coin would have to be done under conditions of isolated environmental conditions such as humidity and air temp not changing, and the wheel spun with identical initial force.

The roulette wheel scenario is orders of magnititude easier to work with beause there are only a few physical variables to deal with. Obviously if you impart a fixed rotational force to a stationary object it's going to spin only N times. Obviously the bearing in this case is the biggest variable since it's going to impart some wobble, heating up due to friction, etc., but I can see this working assuming the initial spin force is very consistent with initial modeling. You also wouldn't need many iterations to produce a predicted outcome.

Side bar about chaos theory, lots of pretty fractals, etc.

 

[BrownTown]

Originally posted by: spikespiegal

As for the Roulette wheels, I have heard that same scenario as well, but its very difficult (much more so than counting cards)

Both the rulette wheel and tossed coin would have to be put in motion with absurd mechanical consistency for countless repetitions for the math to have a chance of working out and producing data for a virtual model. The tossed coin would have to be done under conditions of isolated environmental conditions such as humidity and air temp not changing, and the wheel spun with identical initial force.

The roulette wheel scenario is orders of magnititude easier to work with beause there are only a few physical variables to deal with. Obviously if you impart a fixed rotational force to a stationary object it's going to spin only N times. Obviously the bearing in this case is the biggest variable since it's going to impart some wobble, heating up due to friction, etc., but I can see this working assuming the initial spin force is very consistent with initial modeling. You also wouldn't need many iterations to produce a predicted outcome.

Side bar about chaos theory, lots of pretty fractals, etc.

The roulette wheel thing isn't some theory here, people do it in REAL life, and it does work. The variations are small, but not as small as you might think, certain numbers can be serveral percent more likely to show up than other numbers.

 

[CycloWizard]

If it has a probability of over 50 of landing on one side, I'd say your calculator is broken.

To answer your question, no, there is no way that you can imply the shape of an object from a simple probability measurement. One might be able to model the flipping of a coin with various engravings and thereby predict the probability associated with a given shape, but the reverse is not true. There are infinitely many shapes that could give rise to any one probability distribution.

 

[NanoStuff]

Originally posted by: PolymerTim
Actually, I saw on TV once some people who put this very concept into action. The case concerns roulette wheels and the fact that, as they are used, the bearing they spin on wears down slightly unevenly due to the uneven force with which they are spun.

I heard about that, but this would be the very opposite of that. That is, retrieving the shape of the roulette wheel just from the probability rather than using the roulette wheel to get the probability.

Originally posted by: CycloWizard
To answer your question, no, there is no way that you can imply the shape of an object from a simple probability measurement.

I agree, but just how much can the data be reduced yet still provide sufficient information?

Perhaps the coin can be dropped from three different points of rotation and be timed in how long it takes to fall flat, then, maybe, the weight bias can be triangulated. A top heavy coin would fall faster, and another axis could determine the radial weight distribution of another axis by determining the change in fall rate. Extremely large statistical data should then be enough to be unique to a particular coin face short of slight atomic-scale noise.

 

[BrownTown]

Wait, you need to clarify the question here, are you asking whether you can determine if ANYTHING is on one side, or if you can determine that difference between a picture of a cat verse a picture of a person? The way I interpreted it you were just asking if you could determine if the coin was fair or not, the way PolymerTim is doing it is whther or nto you can determine the shape of the irregularity.

 

[NanoStuff]

Originally posted by: BrownTown
the way PolymerTim is doing it is whther or nto you can determine the shape of the irregularity.

Yes, that's the question The alternative is too mundane and probably obvious.

 

[BrownTown]

Well in that case then you are pretty screwed, but I guess if you did like billions of tests and launched at predefined angles, preferably in a vacuum and stuff then you should in *theory* be able to get some representation of the defect imo.

 

[PolymerTim]

Ahh, that is a little different than I assumed in my previous post. I saw multiple possible meanings in your original post and chose the one I thought was most reasonable. My assumption was that the computer has prior knowledge of the probability of each coin it could be presented with (and that each coin had a distinct probability). That made things easy, but I see your point now.

I think you have clarified and even morphed the problem into a different one. I think Cyclowizard has answered your original question that simply flipping the coin and calculating the probability that it lands on each side is not enough information to detemine what I will call the exact mass distribution (flat, bulging, or ethed patterns included) in 3D. Now you bring up the idea of a more thorough analysis, I think it may be possible, but I'm not sure.

To rephrase your new question, I think you are asking if we can determine the exact mass distribution in 3D given only information about how the coin reacts to a series of specifically imposed forces, such as dropping at an angle and measuring the way it bounces. I'm guessing your trying to basically rule out any visual data.

To be honest, I think you're getting closer, but even in a perfect environment, this would be difficult at best. In a real world environment, I don't think you would be able to design this kind of experiment accurately enough to determine the exact location of scratches in the surface. The first significant problems that come to mind are the nature of the release (when dropping) and knowledge of the exact shape of the coin edge, which will change with each impact during bouncing.

Out of curiosity, what prompted this question. Is this purely theoretical, or is there a real life application that we could push the question towards?

 

[NanoStuff]

Originally posted by: PolymerTim
Out of curiosity, what prompted this question. Is this purely theoretical, or is there a real life application that we could push the question towards?

Just bizarre curiosity for computational inference

I'm sure there is a real life application. A monstrous factoring machine that can compute such a complexity class could have any number of incredible uses with it's predictive power.

The question simply suggests that an object or perhaps other physical system with a finite number of states (undoubtedly astronomical) has a certain unique 'signature', or a hash that either only represents that state or at worst a very limited subset of all possible states that can be computationally recovered by analyzing just that value, or again, maybe a limited set of values. There's probably a mathematical proof that shows how reducible a given state can be without losing information.

At the very least, a real life application could be 'quantum compression'... maybe.

 

[Cogman]

I think that (with that many calculations) One of the possible patterns generated might be a beaver. However, I think that several shapes could give the exact same results as a beaver on one side (even to very high degrees of precision) so it would be impossible to determine which one is which.

 

[Syzygies]

Originally posted by: Cogman
However, I think that several shapes could give the exact same results as a beaver on one side (even to very high degrees of precision) so it would be impossible to determine which one is which.

There's a famous math paper "Can you hear the shape of a drum?" on this same theme. Roughly, the observation space here is a projection of the full shape space, so, no. You'd have more information if you recorded the phase angle of the coin, too, but it would be akin to tomography to reverse this into shape information, and you still wouldn't have enough information. Different shapes would alias to the same probability profile.

There are different ways your mechanic can balance your car tires with the same effect. Same idea.

I'm one of the authors of the "seven shuffles suffice" riffle shuffling paper, so I've heard my share of casino stories. Less common than blackjack card counting is "card sequencing", where you spot a section of the multi-deck shoe that was poorly shuffled, so with a good memory (no better than a competitive bridge player) you know the next card with 25% certainty for a nice stretch. Counting runs on vapors of a few percent and still makes money; a team can really clean up using sequencing. The usual: The big better is a big guy with gold chains and babes on each arm who can really hold his liquor, playing a set strategy he can manage in a coma. The house keeps pouring for him. Upwind, a low-betting teammate passes on cards that the big guy needs, under the assumption that the 25% guess is in fact correct. Upwind from the dealer, a teammate passes on cards that the dealer really, really doesn't need. And a little old lady from Wichita starts winning because the dealer's going bust so often, just thinks it's her lucky day.

 

[dkozloski]

There has been some very interesting work done on determining the shapes of tumbling objects in space by analyzing the glints, nulls, and reflections seen on radar data.

 

[f95toli]

Originally posted by: BrownTown
The roulette wheel thing isn't some theory here, people do it in REAL life, and it does work. The variations are small, but not as small as you might think, certain numbers can be serveral percent more likely to show up than other numbers.

Not any more. Nowadays the casinos "re-balance" their roulette wheels on a regular basis to prevent this, if this is done often enough there simply is not enough time to first gather enough data, analyse it, and then gamble.
But a few people did make a lot of money using this method some 30 years or so ago (I might be wrong, but I think one of them is now a professor of mathematics, at the time he was a PhD student).

 

[Rudy Toody]

Here is a link to a book that describes an electronic attack on the roulette tables.

 

[Smilin]

If you are talking about enough flips to account for the minor variation that a coins imprint can cause you've gotten into the realm where you need to consider one other possibility: The coin may land on it's edge.

 

[gsellis]

Originally posted by: NanoStuff
Theoretical scenario with no statistical noise where a balanced coin would have, to a diminished degree of bias, a 50/50 chance of landing on a particular side and made so that it cannot land in limbo.

The coin however has a certain imprint, say, the shape of a beaver lightly embedded on one of it's sides, giving it a very slight bias to one side. The material is unchanged, the object is simply carved in. The coin is flipped a ridiculous number of times and the probability is recorded as an extremely long floating point value. 50.000002982482...... so on.

Assuming a powerful enough computer(absurdly powerful mind you) that simulates flips of a countless number of differently shaped coins:

After being fed in a an extremely precise probability number, can the computer determine, to a high degree of precision, that the coin with this particular probability has a beaver engraved on one of it's sides?

Your computer is biased by the way. It has problems with true random generation. It may take it longer to get a real random result than to flip an actual coin.

As for the beaver coin, even a long trial is not going to give you an exact result. The sample size should reduce the variance, but there is still a probability that you should get a deviation from the expected result.

Remember, in math 2+2=4. In statistics, it is improbably that 2 + 2 is something other than 4. Huge difference in theory

 

[firewolfsm]

It should be able to determine the bias, but there are an infinite amount of combinations of depth and shape that could produce that bias, not just a beaver.

 

[Stiganator]

I don't think you could figure out that it is specifically a beaver. You would not be able to differentiate the beaver coin apart from something with the same moment (including drag and such) when thrown, assuming that the moments are not somehow unique. The statistics you are looking at wouldn't provide enough data to reassemble the geometry. That's like saying, my wallet weighs exactly 45 grams, there must be $500 in my checking account. You can think of a statistic as "lossy". It is a simplification of all the data that you could get from the system. Now, take an infinite number of statistics and you could tell most certainly.