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23rd February 2011, 08:49 PM | #121 |
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If was? Because that's not what you said, and it's not what everyone here is responding to.
The odds that you'd ever actually encounter a run of 100 heads or tails in a fair coin-flipping system are exactly the same as the odds of encountering any other unique string of 100 heads and tails. That's true by definition of "fair". That is what everyone is trying to explain to you. Yes, those odds are very very tiny - but it's quite literally nonsense to assert they are zero, as has now be explained ad nauseum. |
23rd February 2011, 08:58 PM | #122 |
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Ah, you're right, I never meant that it was possible to throw 100 heads in a row on earth, I always thought you were talking about doing it on Mars. On earth (where physics and probability may work differently from the rest of the universe..?), maybe it's not possible. Huh?? |
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23rd February 2011, 09:14 PM | #123 |
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Sure, and to do that you look at the actual properties of the system, our understanding of the physical laws of the universe, and deduce the implications of that.
So, for instance, in the case of coins, as sol pointed out there are no fair coins, but we can figure out the probabilities of coming up heads vs. tails. We can then deduce (from the fact that neither is impossible and that subsequent flips are in no way influenced by past flips) that all combinations are possible. If we use the abstraction of a fair coin, that's fine, and in that case we find that all combinations are equally probable. It really is that simple, and it really is based on the actual physical system involved. |
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23rd February 2011, 09:39 PM | #124 |
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Piggy's viewpoint here seems to me to be increadibly unscientific, from two perspectives.
The first, and most important, is the view that we can't know anything if we haven't done that specific experiment. It comes down to something like this: we've measured the speed of light to come out to approximately 3x108 m/s. There are some valid questions about whether or not that value has changed with time or is different in different parts of the universe. But Piggy's view is like asserting that we don't know that speed of light is on sunny wednesday afternoons when the temperature is exactly 10.005oC and the finance minister of Kenya has just given a live speech for French television, because we haven't done that experiment. When asked how those things could possibly affect the speed of light he'll reply that he doesn't know, but you can't possibly know either. Now, that may be true in a way, but I'm confident that if we do measure it in those circumstances it will come out to be the same as it was that last time we made the measurement. And if we refuse to make those sorts of assumptions, science becomes basically useless, because we can no longer put confidence in any conclusions. The point of science is to make useful predictions, but they are predictions about a universe that is different than it was at any time when previous experiments were made. The second problem I have is that Piggy actually goes further than this: he accepts the abstraction of a fair coin, from which the conclusion that any possible combination is equally likely necessarily follows, and then goes on to suggest that the messiness of the world will make the coin not fair (not in so many words, but that's what his argument amounts to) but doesn't realise that he has just changed the scenario by doing so. Either we can discuss fair coins (in which case what colour underwear Tom Cruise is wearing today doesn't affect the results of our coin tosses), or we can't. But it's silly to talk about fair coins and then suggest that there are outside forces that affect their results - if those outside forces exist the coin isn't fair. |
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23rd February 2011, 09:41 PM | #125 |
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I don't think that's right. The machine doesn't remember the results of previous flips and change future flips based on them. It has no mechanism by which to do that. Flips are independent. It's just that the machine can be set up so that, on each flip, the probability of getting heads is 100%.
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23rd February 2011, 11:29 PM | #126 |
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Here's one way a fair random binary bit generator (equivalent to a fair coin flipper) that was rigged not to permit any runs longer than 25 might be detected, in the real world. You own a chain of 10 casinos. Each casino has 500 one-dollar slot machines in it of a certain type, each of which gets 2000 plays per day. That type of machine pays (among the rest of its payout schedule) a jackpot of 5 million dollars, with the odds of winning the jackpot on a one-dollar bet set at 1 in 67.109 million. The odds of generating 26 1's in a row (i.e. all "heads") from a fair random binary bit generator are 1 in 67.109 million, so the Gaming Control Board okays a machine design based on using one of those as a randomizer and scoring a jackpot win whenever 26 1's come out. They do not know that the particular circuit you use in the machines has been rigged to prevent any runs of more than 25 1's. After all, it's impossible (or so you think) to tell the difference. And yet, after only two weeks, when no jackpot has been won despite 140 million plays, the authorities start to get suspicious. After four more weeks, they launch a full investigation. Shortly thereafter, you go to jail, unless you've already fled the country with the money. Respectfully, Myriad |
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23rd February 2011, 11:34 PM | #127 |
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24th February 2011, 01:37 AM | #128 |
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Unlikely. Only about 1 in 2^100 runs will show 100 heads, so divide the number of runs you 'encounter' by that number and you get an idea of the likelihood that you encounter a run of 100 heads.
Now perform 100 fair flips and note the sequence. That sequence had the exact same likelihood of appearing as 100 heads. Yet, it happened, so it can't be impossible. |
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24th February 2011, 02:43 AM | #129 |
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24th February 2011, 08:46 AM | #130 |
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Here's the important point. You made a claim - that a run of 100 heads is not possible. You have no evidence to support that claim, since the only evidence that you haven't seen one also supports the null hypothesis. You also have now made it clear that you have no theory to suggest how it could possibly be the case. That's pretty much a textbook definition of blind faith.
Edit: This is the point I was making with the monkeys as well. As long as there is a finite chance for any key to be pressed each time, you are guaranteed to eventually get the works of Shakespeare. It's exactly the same situation. As long as the scenario isn't artificially constrained, such as the biased generator above preventing runs of more than 25, changing the weightings only changes the exact probability, not the qualitative result. |
24th February 2011, 06:10 PM | #131 |
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Well, the OP isn't mine, btw. And there's more than one train of conversation going on, as usual.
Originally, I was talking about whether you'd ever get a series of 100 heads or tails by flipping a coin (with your hand) here on Earth -- even assuming that the task of flipping might have to be handed on from person to person. Then I got into a conversation with Ivor about the possibility of designing a fair coin-tossing system (which a human may or may not be) that could be fair and limit streaks. And by that time there were other posters discussing other things. Anyway, by now I've had to turn tail on the "can't get to 100" idea. That didn't pan out too well. But there's still no way to know whether human coin-flipping actually is fair or actually does allow non-bounded runs. And I'm certain that it's possible to create a system which is indistinguishable by humans from a random system, but which excludes streaks beyond a finite boundary. |
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24th February 2011, 06:11 PM | #132 |
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24th February 2011, 06:14 PM | #133 |
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24th February 2011, 06:14 PM | #134 |
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24th February 2011, 06:18 PM | #135 |
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Actually, there is:
Originally Posted by me
Quote:
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24th February 2011, 06:28 PM | #136 |
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24th February 2011, 06:29 PM | #137 |
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My understanding is that if you're dealing with the results space of a random system, it's literally impossible to know what it will be until the calculations are actually performed.
The speed of light in a vacuum isn't comparable, because it is a constant (as far as we know). On the other hand, very different sorts of real-world values, such as the price of oil, do indeed depend on weather, politics, and media. So if we're dealing with a brain controlling a hand, what sort of system are we looking at, precisely? What will its results space look like when it comes to coin-tossing? Is there anything in that system that will limit streaks, such as unconscious sabotage, for example? Is it truly random, or is there actually a very few number of states that don't vary perfectly randomly? I don't believe answers to those questions are yet available. I'd never suggest that for the speed of light in a vacuum, but for a brain and a hand, you bet. What measurement? You've never made any measurements of people flipping coins that would allow us to draw conclusions about whether it actually does run through all possible combinations or not. Neither has anyone else. No, it's actually good science not to overgeneralize or overstate. Look at the decelerating universe or the ever-expanding black hole. We don't know enough about people flipping coins to say if it's true that all mathematically describable combinations will actually be demonstrated, given enough time. Why is that so difficult to accept? Actually, this was the point I concede to Ivor. You can't have a fair coin that limits streaks. The two are incompatible by definition. But you can certainly have a coin that is "locally fair" in that its behavior is literally indistinguishable from the behavior of a fair coin by the people using it, even though it limits streaks. |
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24th February 2011, 06:31 PM | #138 |
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24th February 2011, 06:34 PM | #139 |
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24th February 2011, 06:35 PM | #140 |
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24th February 2011, 06:35 PM | #141 |
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There are a few problems here: one is that to limit streaks requires a memory: now certainly a human brain has a memory, but in order to be effective it has to actually be able to affect the outcome of the toss. So the human brain has to have a tendency to get the hand to flip the coin in a way that it will come up tails when there are too many heads in a row. I can't control a coin toss when I want to. You are suggesting that people just naturally do this unconsciously.
Furthermore, they have to be able to do so flawlessly. If their control is less than 100% there will still be a chance of getting heads, and that means that while it's less likely because of this mechanism, it's still possible, and will happen with enough iterations.
Quote:
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24th February 2011, 06:40 PM | #142 |
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Oh, no. I've already said, the results-space appears to be infinite.
So I can't say that 100 heads would never come up. But by the same token, I can't assert that they're possible, because I don't know whether or not there's something in the system that could limit them. It's possible, for instance, that any given human's arms and hands will exhibit only a small number of actual configurations when flipping coins, so the pool of results is much smaller than we might think. And it's possible that the person's understanding of the previous results might somehow influence which one of those configurations will be executed. I'm not saying that this actually occurs. All I'm saying is that I don't know enough information about the system to conclude that it will in fact run through every mathematically possible combination of states. |
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24th February 2011, 06:43 PM | #143 |
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24th February 2011, 06:45 PM | #144 |
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24th February 2011, 06:48 PM | #145 |
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No, you're not guaranteed that at all.
Because the question is actually one of predictability. It may be impossible to predict, by looking at one keypress, which key will be pressed next, but still to be looking at a system which will never produce a single work of Shakespeare. |
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24th February 2011, 06:53 PM | #146 |
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The same laws of physics apply, but the set-up is not identical.
I don't think that the difference will affect coin-tossing, but it can certainly affect other systems we might want to measure, and I can't absolutely swear that it won't affect the results you get from people flipping coins. It would certainly be possible to rig a 2-state Diaconis machine to produce results that are indistinguishable from random by humans, while limiting the absolute length of streaks. And unless you claim to know all the relevant influences in a system comprised of a human and a coin, you simply cannot assert that the results-space of this system will include all mathematically describable combinations. |
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24th February 2011, 06:55 PM | #147 |
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24th February 2011, 07:02 PM | #148 |
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Let's keep some perspective here.
I conceded that my notions about streaks of 100 not being possible were indefensible. Not because I thought a streak of 100 likely, but because I had to concede that I can't find any absolute barrier to it. By the same token, if you really do want to assert that streaks of 100 actually are possible in a particular system then you have to demonstrate why this is so. I cannot say that streaks of 100 are impossible, but I also cannot say that they must be possible, given any particular system of flipping coins. I cannot claim to know that human coin-flipping is in fact unbiased -- no matter how unlikely that might turn out to be, short of zero -- for the same reason that I cannot claim to know that a streak of 100 is impossible. |
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24th February 2011, 07:04 PM | #149 |
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24th February 2011, 07:28 PM | #150 |
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You can set the machine up so that whichever way the coin is facing before it gets flipped, it will face the same way after it gets flipped. So set up, the machine, in a sense, remembers which way you put the coin in. But it doesn't remember the results of previous flips. Those results have no effect on the current flip. If the current flip results in heads, that's because you put the coin in heads up this time; it has nothing to do with what happened last time. That's what I meant.
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24th February 2011, 07:36 PM | #151 |
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It's easy to believe that it might be slightly biased---e.g., perhaps the probability of heads is 53% instead of 50%---but it's very hard to believe, at least for me it is, that after getting 99 heads in row, the next flip is absolutely certain to come up tails. I mean, there's nothing special about that particular flip. Presumably, whoever's flipping the coin flips it more or less the same every time. If all the other times, the probability is roughly 50:50, why not this time too?
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24th February 2011, 07:40 PM | #152 |
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24th February 2011, 07:46 PM | #153 |
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Is it any more difficult to believe that a brain-hand system might actually limit streaks than it is to believe that 100 consecutive heads might actually turn up?
In other words, if we're talking about what can't be proven impossible, then certainly 100 consecutive heads shares the same greenroom with a streak-limiting human brain-body. |
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24th February 2011, 07:49 PM | #154 |
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Cool.
Quote:
So, for a streak (however long) to be impossible, at some point it has to be impossible for a particular flip to result in heads. That is counter to the reality of the system: it is never impossible for a human coin flipper to flip a coin and get heads. If you don't disagree with that then the issue seems closed.
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24th February 2011, 07:56 PM | #155 |
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Actually we do, because we know what the components of the system are. The only component of that system that has a memory is the human brain. And we also know that it is not capable of unconscious and perfect control of the flipping. So, even if we make the somewhat odd and un-evidenced assumption that humans tend to unconsciously affect their coin flipping in such a way as to limit streaks, they are still incapable of doing so perfectly, and thus all streaks are still possible.
Furthermore, if this is your argument it seems that you concede that if, for instance, the human flipper were unaware of the outcome of the coin tosses (and thus any possible memory were taken out of the system) that all sequences would be possible, correct? |
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24th February 2011, 07:57 PM | #156 |
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24th February 2011, 09:13 PM | #157 |
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It's not at all hard for me to believe that the chances of getting 100 heads in a row are about 1 in 2100. In fact, that's exactly what I believe. 1 / 2100 is very small, to be sure, but it's not zero.
It's much harder for me to believe that the chances are precisely zero, because that would imply that the 100th flip is qualitatively different from the first 99: each of the first 99 might come up heads or tails, but if they all happen to come up heads, then somehow the 100th flip inexplicably must come up tails. What could possibly be special about that particular flip, compared to all the others that have ever taken place in the history of the world, which makes it certain to come up tails? |
24th February 2011, 11:00 PM | #158 |
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25th February 2011, 12:11 AM | #159 |
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It's been mentioned a few times, and I am baffled that the thread didn't stop dead...non-zero probability means possible. Why would not having perfect knowledge of a particular system mean anything, unless it somehow affected the probability of a given set of outcomes?
Is a given sequence possible? What's the probability of that? Oh, it's a real number between 0 and 1 that happens to not be zero? Then it is possible. |
25th February 2011, 01:19 AM | #160 |
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Why did you remove the second half of that quote and then type something that is contrary to the removed part:
So even if your "human brain/arm/hand" idea reduced the chance of heads to 1%, any run of heads would still be possible. The only way your theory would work is if somehow the "human brain/arm/hand" idea reduced the heads possibility for a particular flip to zero. Good. |
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