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15th February 2011, 07:14 AM | #41 |
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You don't have to cheat if your opponent has no grasp of probability:
http://www.scribd.com/doc/13366448/T...lay-in-One-Act |
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15th February 2011, 07:31 AM | #42 |
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Any particular combination will be a statistical long shot. Some combinations are statistically significant: they are better explained given cheating or bias than chance alone.
As Piggy stated, you'd be a fool to bet a coin that's landed heads 100 times in a row will come up tails. This is because you intuitively know that that many heads in a row is far more likely given a two-headed coin (or other form of cheating) than by chance. |
15th February 2011, 07:34 AM | #43 |
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No you haven't. If you have, it's prima facie evidence the die is badly weighted.
That's a 1 in 100,000,000,000,000,000,000 event, or one in one hundred quintillion. To answer the OP, after 5 in a row, I'd say, "Hold on. I want you to do 10 test flips, please." If your opponent seems a touch unsavory, demand it before you start. Of course, a decent prestidigitator could easily swap the fraud one in and out. Quite frankly, I don't even know if it's possible to weight a coin so it lands on one side more than the other more than a fractional increase. Even the best would be pushing it to do 10 in a row, I'm guesstimating. I'll also add that it will probably bounce in ways that look funny. |
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15th February 2011, 07:51 AM | #44 |
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Could you clarify this a bit, as far as I know, and based on my experience, unless we are trained, our brains are awful statisticians.
If you want me to cite evidence: Las Vegas If that's too artificial for you: Prayer, Homeopathy, Lucky Charms (not the cereal). |
15th February 2011, 05:59 PM | #45 |
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That was one of the things I had hoped to bring out in the paper - the difference between the skeptic and the believer is represented by differences in the prior. The trouble with irrational belief is that it does not change with evidence. In the coin-toss example, an irrational belief would be represented by a prior of 1 or 0. I probably need a more detailed section on investigation... I was kinda hoping that the discussion here could guide me as to what I needed to talk about ... like I've probably been too pedantic in some bits and too vague in others. While we quibble over confidence thresholds, the effect of the prior makes the exact threshold unimportant - technically the usual fight is over what constitutes evidence. The MDC experience shows us that it is very difficult to set up an unambiguous trial.
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The actual analysis sets up a false dichotomy by only taking into account the possibility of cheating by double-headed-coin... though what I really ant to do is eliminate fair chance as a cause for the observed results.
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What I am bringing up in the article is probably a small point for skeptics, but one that is often misunderstood by, um, most people. Since a great deal is popularly written on the points you have brought up, I thought I'd concentrate on this one. So - good putting the paper in context of the broader field, but I'm not writing a book here (later?) I think the math works better if the model is of many coins in a bag, a certain proportion of which are double-headed. I draw one coin from the bag and start tossing. It also makes the language a bit more neutral. |
15th February 2011, 06:31 PM | #46 |
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re: run of 10x00 result on d100:
However, the same calculation holds for any particular sequence of numbers rolled. Why is this particular one so special? Of course, in D&D, the 00 roll usually indicates a failure (you have to roll d100 [i]less than[/] the number to succeed or it's roll generating percentages does not work) ... whatever, this number is special if anything else has another meaning ... 1% chance of failure and 99% otherwise. So a long run on failures would be a remarkable event. Note: in D&D irregularities in the dice are part of the game ... players agree to use a particular set for the game, and this is fair because all players have access to the same dice - and so the same bias.
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Can you quantify your confidence though? If you saw 4 heads in a row would your reaction be different? How much by? You are guessing right? Why should anyone believe your guess as opposed to someone who wants to be suspicious after two flips, or twenty? Your opinion, though, agrees quite well with the calculations if you think the chance a cheater is involved is about 1% ... but if you want to reject it outright at the 5th toss, then that would be rational if you give equal weight to the possibility of cheating. Have you read the article?
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15th February 2011, 06:35 PM | #47 |
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I was hoping someone would check the math ... do I take it that everyone who has read the paper agrees with the math?
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15th February 2011, 06:43 PM | #48 |
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It's a little glib isn't it? He's talking about the kinds of stuff we are evolved to deal with - like spotting faces against a random background: we are good at it - a bit too good - but if you consider how often we detect a pattern that really is there (reading this post, identifying a photograph for eg) against the false positives (spotting BVM/elvis/spacecraft in a cloud), the odds are pretty good. That's not exactly doing statistics (what are the odds a particular influx of light on the retina represents something I need to pay attention to?) any more than catching a ball is exactly doing differential equations.
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15th February 2011, 07:13 PM | #49 |
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Because it disconfirms, to an extremely high degree, the null hypothesis of random chance. To put it another way, there are many, many combinations that are consistent with random chance. However, the set of combinations that is consistent with cheating is much smaller. So when we see a result that belongs to the set that is consistent with cheating, a bunch of red flags go off.
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15th February 2011, 08:31 PM | #50 |
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I used to teach test prep for exams like the SAT, GRE, LSAT, and such, and I can tell you that the human brain is indeed very bad at statistics when it comes to abstract problems, and situations -- such as the shared birthday question -- that are simply of no importance in everyday real life.
Vegas, of course, intentionally manipulates the casino set-up to exploit that fact. I don't think that prayer and homeopathy are believed in because of a failure to comprehend the stats intuitively... seems to me there are other psychological blind spots at work there. But consider that even babies who have not yet learned to talk are cracker-jacks at the kind of stats that are important to us in everyday life. Babies will stare measurably longer at unexpected events than they will at expected ones, and we can use that fact to judge when they sense that something is fishy. If a baby is looking at a clear plastic box filled with red and white balls, and the white ones greatly outnumber the red ones, for instance, and an adult removes several red balls in a row from the box, the baby shows perplexity at this situation. S/he knows something is wrong with that. Also, we know that babies will attend longer to novel sounds. But only recently has it been discovered that when babies are exposed to various pairings of sounds, it doesn't take long for them to learn which pairings are common and which pairings are rare, and to respond with increased attention when a rare pairing occurs with an unexpectedly high frequency. Turns out, our brains are naturally built for statistics, but only within the realm of what biology has determined is particularly useful. This is why we're flummoxed by the birthday problem, but we will refuse to bet on a coin that comes up heads too often in a row, because we know the real universe does not work that way. If you've got a fair coin and a fair set-up -- normal atmosphere, a human hand doing the flipping, etc. -- then the actual randomization of the world we live in sees to it that 100 heads in a row never actually happens. Yes, it's possible on paper, but not in our actual universe. |
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15th February 2011, 08:50 PM | #51 |
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So it's impossible?
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15th February 2011, 09:56 PM | #52 |
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15th February 2011, 10:06 PM | #53 |
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2^100 is about 10^30, which is about how many viruses there are on earth at any given moment. So if there's some improbable event that applies to viruses and has probability equal to flipping tails 100 times on a fair coin, it happens to one every day (or however long viruses live).
Therefore it's not correct to say such events are impossible. If you want a cutoff on probabilities below which you can say the event never happens, it needs to be much smaller than that. And anyway, as you know all sequences of flips are equally improbable. Since coins have been flipped far more than 100 times in the history of the world, the sequence of all coin flips that so far have taken place has a probability that's 2^{-N}<<2^{-100}, where N is the number of flips.... and yet that event (flipping that sequence) happened. |
15th February 2011, 10:09 PM | #54 |
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I'm trying to see why you think that, Piggy. I think it's just a loose definition of "impossible", but I'll ask this just in case:
If you toss a coin and in comes up heads, is it more likely to come up tails the next time? And the time after that? |
15th February 2011, 10:11 PM | #55 |
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I don't live on paper.
I live in the real world of weight and heft and air currents and muscles and neurons. And in this world, a fair coin will not come up heads or tails on 100 consecutive fair flips. We may not understand exactly why our world works that way, but it does. |
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15th February 2011, 10:11 PM | #56 |
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15th February 2011, 10:16 PM | #57 |
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That's the wrong question to ask. It's looking at the landscape through a soda straw.
Yes, the coin is stateless. It doesn't know what happened on the previous flip. The pertinent question is: What will happen if a human being flips a fair coin 100 times here on the surface of the earth? What we observe is that runs of 100 do not occur. In fact, nothing even close occurs. And that's because the physics of the coin and the atmosphere and gravity and human muscles ensures a certain level of random variation. As I said, we may not yet be able to explain why this is so, but there's no doubt that it is so. And referring to a fictional, abstract, pen-and-paper world of pure statistics doesn't change that fact, because we do not live in that world. |
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15th February 2011, 10:17 PM | #58 |
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15th February 2011, 10:19 PM | #59 |
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15th February 2011, 10:22 PM | #60 |
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Do you recognize the irony in your position? You claim to know that sequences of 100 coin flips never come up all tails in the "real world", and that we're talking about some kind of detached-from-reality theory world.
But how many times have you flipped a coin 100 times in a row? Obviously not enough times to conclude any such thing - and therefore, you're the one using theory. And using it incorrectly, I might add. |
15th February 2011, 10:24 PM | #61 |
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It's not incorrect, at least not unless the flipper starts with the coin with heads or tails up completely randomly. But since the whole point of the coin flip is to generate a random heads or tails, that would be rather circular.
It's essentially impossible to flip a coin "fairly", due to conservation of angular momentum and the mechanics of coin flipping. |
15th February 2011, 10:24 PM | #62 |
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"Fair" in this sense means a coin with an equal chance of landing on heads or tails.
http://en.wikipedia.org/wiki/Fair_coin I do understand that things in real life are different from things in theory when things like perfection (such as a fair coin) are involved in the theory, but that this does not make it impossible for unlikely things like this to occur.. |
15th February 2011, 10:27 PM | #63 |
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15th February 2011, 10:35 PM | #64 |
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Yes, I am aware of that.
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15th February 2011, 10:35 PM | #65 |
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As a gambling man, I've dealt cards, rolled dice, and tossed coins enough to know how the patterns go down.
Not only that, but I've observed other games, heard from other gamers. In this world, runs of 100 heads or tails on a fair coin don't happen. Your numbers on paper don't change that. And I can be content with not understanding precisely why the physics of the world I live in give us those results, while accepting that they do. Your calculations change nothing. |
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15th February 2011, 10:36 PM | #66 |
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15th February 2011, 10:40 PM | #67 |
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And viruses, I might add. As sol said, given enough iterations we can be fairly confident that a series of 100 consecutive heads will turn up. And in the real world there are things that do have enough iterations of basically random events that these sorts of things can and in fact do occur.
If, for instance, we were to bet that a bacteria would never evolve along a particular molecular pathway because the chances of the mutation were 1/1030, there's a good chance that we'd turn out to be wrong. And that's the real world. |
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15th February 2011, 10:44 PM | #68 |
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Sol's calculations predict exactly the same outcome in your games and those of other gamers as you do. He doesn't expect you to have a run of 100 heads either. So clearly you can't distinguish your theory (that it's impossible) from his on that grounds.
On the other hand, his viewpoint predicts that in the case of bacteria that I supplied in my previous post, in a large enough population the mutation would arise. Your viewpoint necessarily predicts that it won't, as something that's impossible in one iteration is necessarily impossible in the second, third and 1030th. |
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15th February 2011, 10:45 PM | #69 |
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15th February 2011, 10:55 PM | #70 |
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Why do you believe that, when experience tells you otherwise?
In the world I live in, the randomness built into our universe restricts such runs to much smaller ranges. Abstract mathematical worlds that round off all the edges and don't account for the complexity of reality are not, as far as I can see, adequate descriptors, so I do not defer to them. |
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15th February 2011, 10:59 PM | #71 |
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Sure I do. My brain is built for this world, and is in fine working shape.
I don't yet know, because nobody knows, precisely why the physics of our world results in the actual behavior we observe. But that's no reason for me to abandon actual experience for abstractions which describe idealized worlds stripped of all the stuff we don't yet have a handle on. To accept that world as the true world would be insanity. They are not "fair" according to a description which, as far as I can tell, is not particularly relevant to reality. |
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15th February 2011, 10:59 PM | #72 |
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When you said it was impossible, I felt like poking fun of that position by suggesting that you think that it would be against the laws of physics. I didn't do that, because I thought that would not be fair.
But it seems you do think it's against the laws of physics. "Patterns"? Does that mean you can predict what will happen? You are arguing against simple matters of mathematics, based on anecdotes. |
15th February 2011, 11:03 PM | #73 |
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For much the same reason that I think that if a star has a high enough mass, when it goes supernova a black hole will result: because we have a theory about what is happening, a theory that is increadibly robust and has stood up to all attempts to falsify it, and that is what it predicts.
Similarly our understanding of mathematics and probability, along with the physics of coins, is very well tested and understood, and it predicts that given enough iterations such a string would occur. But here's the thing, if we could find an event that did have enough iterations that I would expect something with an equally low probability as 100 heads in a row to happen, would you expect that such an event would not happen? |
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15th February 2011, 11:06 PM | #74 |
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There's a very large opaque bag with a billion balls in it. One of those balls is red and the rest are white. I remove some balls without looking until they are removed, see that they are white, and then I insist that there is some mysterious force of physics that is forcing me to pull out only white and that there's no way I could possibly pull out a red one.
I think that is similar to this. |
15th February 2011, 11:07 PM | #75 |
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The difference here is that observation confirms the black hole theory, but not the "100 heads on a fair coin is possible" hypothesis.
Our universe behaves in a way consistent with black hole theory. It does not behave, as far as we can tell, in a way consistent with the 100-heads hypothesis. |
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15th February 2011, 11:08 PM | #76 |
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15th February 2011, 11:14 PM | #77 |
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We've never actually observed a black hole forming. So, no, we don't have direct observational evidence that a large enough mass star will form a black hole. We only have the consequences of very well tested theory.
Quote:
Please note that there is no prediction that "100-heads"* are commonplace, only that they happen when there are enough iterations. *And also note that I am talking about all events with that probability. |
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15th February 2011, 11:15 PM | #78 |
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And yet, that's precisely what you are doing.
Quote:
1) start with a normal coin on your cocked thumb, heads up. 2) flip it normally 3) catch it, and (say) don't turn it over 4) look at it You'll find that it's heads measurably more often than it's tails. That's as it should be based on Newtonian physics, and it's been confirmed experimentally. Not only that, people can learn to control the effect in a way so that the flip looks normal, but the result is much more likely to be heads than tails. ETA - and the same goes if you exchange "heads" for "tails" everywhere in that post (what matters is which side starts facing up). |
15th February 2011, 11:16 PM | #79 |
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Actually that seems to sum up your evidence pretty well. Where does the comparison break down?
Or to make myself more clear, you accept that a string of two heads is possible. What number of consecutive heads is the maximum possible number? After that number has been reached, what do you think stops another head from turning up? |
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15th February 2011, 11:20 PM | #80 |
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I'll live in my world, y'all live in yours.
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