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21st February 2011, 07:18 PM | #81 |
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21st February 2011, 07:19 PM | #82 |
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Progress!
Do you think that it is less likely to come up than any other single combination? |
21st February 2011, 07:21 PM | #83 |
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21st February 2011, 08:43 PM | #84 |
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You're making this vastly more complex than it really is.
If you would ever get around the answering my question, you might understand why. It's got to do with your red/blue thing above - if you (randomly or otherwise) re-color heads and tails on each flip, 100 heads could be an even mix of red and blue, while some more random sequence of heads and tails would be all red or all blue. 100 heads is no more or less special than any other sequence of 100. The only way that can be wrong is if the coin has a memory. |
21st February 2011, 09:10 PM | #85 |
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It is not possible to make a machine that randomises it and limits streaks, by definition. Anything that limits streaks stops it from being random.
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21st February 2011, 10:10 PM | #86 |
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21st February 2011, 10:15 PM | #87 |
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Hmm?
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22nd February 2011, 02:00 AM | #88 |
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The size of the "results space" doesn't matter. It's 50/50 whether a coin comes up heads every time you flip it. The largest cluster can be 100% of all coins flipped (and by that I mean all coins flipped ever, in the entire history of the universe). The largest cluster of heads can be 0. It can be exactly 50/50 split between heads and tails.
None of these results is any more or less likely than any other specific result. |
22nd February 2011, 04:23 AM | #89 |
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So Piggy finally admitted that 100 heads in a row is possible? He just used 918 words to say it so we'd forget he was wrong....
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22nd February 2011, 04:25 AM | #90 |
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22nd February 2011, 04:33 AM | #91 |
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22nd February 2011, 11:04 AM | #92 |
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I like this response, from a testing standpoint. Impossible indicates a prohibition of some kind, as opposed to finite, non-zero probability. If some given number of flips results in some prohibited state, you have to establish why that number is different. Gambler's fallacy, etc and so on.
by the way, enjoying the forum immensely. |
22nd February 2011, 02:55 PM | #93 |
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23rd February 2011, 02:38 AM | #94 |
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Its not impossible, just improbable, it is so improbable that is has seldom happened and will seldom happen.
The average number of tosses to get 100 heads in a row is (2^(100+1))-2 = 2,535,301,200,456,458,802,993,406,410,750 Feel free to try and beat that number. Your chance of getting 100 in a row will be 1 in 2^100 = 1 in 1,267,650,600,228,229,401,496,703,205,376 |
23rd February 2011, 04:35 AM | #95 |
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I have a math query...
Was thinking about this as I was in bed last night- So; I decided I would not continue flipping coins if I hit a tails- instead I would start over again. How would this effect the speed in which I flipped 100 heads? Going by the asumption I can flip and write down the result in 5 seconds. Also would be grateful if you could work out the MAX time taken to flip 10/25/50/75 in a row. I can halve this to find the average I guess. Thank you fellow mathematicians - |
23rd February 2011, 06:05 AM | #96 |
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This thread deals with that question. I think the short answer is that it speeds things up by a factor of 50.
There's no maximum. It could take arbitrarily long. But sufficiently long times are very improbable. No, it's not that simple. |
23rd February 2011, 06:47 PM | #97 |
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I'm talking about the physical system. After all, what other system is there? When we're talking about coin flips, we're necessarily talking about some physical system, whether it's a machine or my right hand.
If you want to say that any particular system has a particular results-space, and not some other, you're going to have to show why that is. It's not enough to demonstrate what all the possible combinations are. You also have to demonstrate that the system will achieve them all. |
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23rd February 2011, 06:49 PM | #98 |
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23rd February 2011, 06:53 PM | #99 |
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Yes. I was referring to a fair coin.
Is a heads more likely to come up than a tails? No. Any result is as likely as any other, whether you have just one flip or any number. |
23rd February 2011, 06:57 PM | #100 |
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In this context, your scenario creates an entirely different question.
The original question concerned how many times a (very crude) measuring device could be applied to a system and end up with a particular result, if the system itself were truly random. If you randomly recolor the results of those measurements, you get a results-space that's the interaction of the two systems. So if you want to answer questions about the first system (the coin flips) you have to reverse the re-coloring you did in the second step. In other words, the re-coloring does not (because it cannot) add anything to our understanding of the first system. If it were true that a run of 100 were impossible in the first system, and it could be deduced from the results-space, the random recoloring would mask that fact. If it were true that the results-space must be monochrome, the random recoloring would mask that, too. As far as I know, we're not asking about the intersection of systems. But back to the issue of a series of coin flips... we should be just as cautious about proclaiming what a physical system can do as we are about proclaiming what it can't do. |
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23rd February 2011, 06:57 PM | #101 |
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23rd February 2011, 06:59 PM | #102 |
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23rd February 2011, 07:01 PM | #103 |
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23rd February 2011, 07:03 PM | #104 |
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What things?
By that I mean, math is very good at describing the probabilities of a system whose operations are exactly known -- e.g., a physical system that's bound to exhibit every possible configuration of its parts in a truly random manner -- but where there are unknowns, math can't reach. |
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23rd February 2011, 07:07 PM | #105 |
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It will be approximately 50/50, as I have explained before.
How do human hands and air and metal and so on change the likelihood of it getting any other sequences? |
23rd February 2011, 07:08 PM | #106 |
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23rd February 2011, 07:09 PM | #107 |
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23rd February 2011, 07:10 PM | #108 |
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The problem with that is the same could be said about every other single combination.
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23rd February 2011, 07:12 PM | #109 |
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23rd February 2011, 07:15 PM | #110 |
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True, but it doesn't stop it from being indistinguishable from randomness from the perspective of the people using it.
Set up a Diaconis device with a computerized switch that excludes runs of, say, over 25, but randomizes the results otherwise, and nobody in our world will be able to detect that it's not truly random. In other words, it will exhibit behavior that cannot be distinguished -- within the given timeframe -- from the behavior of a similar system which does not place any such limits on streaks. That being the case, if the results-spaces are identical from the perspective of the people using the system, I don't see any way to argue that it's not effectively random for the purposes they're using it for. |
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23rd February 2011, 07:16 PM | #111 |
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23rd February 2011, 07:17 PM | #112 |
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23rd February 2011, 07:21 PM | #113 |
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23rd February 2011, 07:45 PM | #114 |
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Math is good at deducing which consequences absolutely must follow from given definitions and assumptions. Definitions are of course true by definition, and whether the assumptions hold in the real world is up to experiment to decide. But if the assumptions do hold, you can be sure that the consequences deduced by math also hold.
Getting back to coins, if on a single flip, a coin might come up heads and it might come up tails, and if also there is nothing in the coin (or the flipping mechanism) that remembers the results of previous coin flips and alters the next flip based on them, then it follows inexorably that 100 heads in a row are possible. So there are two choices: 1) 100 heads in a row are possible, or 2) If you've already flipped 99 heads (or possibly some smaller number), the coin somehow knows this, and will definitely come up tails on the next flip. Now, I think it's reasonable to say that a coin is not being flipped fairly if it will definitely come up tails. So, by definition of "fair", there's a chance that a fair coin which is flipped 100 times will come up heads every time. Perhaps there are no fair coins in the world. You can consistently maintain that position, I suppose. But just be aware of what it implies. |
23rd February 2011, 07:49 PM | #115 |
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It illustrates the absurdity of the claim that 100 head runs are impossible. I was hoping it would hence add something to your understanding of this system - but days later you still haven't responded to my questions about it, so I guess not.
By the way there's another argument that's rather clear. You assert 100 head runs are impossible, p(100)=0. You also agree that 5 head runs are possible, p(5)>0. At some point someone asked you how many heads in a row are needed before it becomes impossible, and you said you doubted there is "a bright line". But there must be a bright line if you are correct: if p(5)>0 and p(100)=0, there must be an N such that p(N)=0 but p(N-1)>0. Perhaps N=100, or perhaps N<100, but there must exist such an N. That means that if we flip N-1 coins often enough, eventually we are certain to get a run of N-1 heads (since p(N-1)>0). And then, if we flip once more, we are absolutely certain to get tails. With probability 1, we must get tails. Our coin is magic - it refuses to be fair, because it cannot allow a run of N heads. And that, of course, is utterly absurd (not to mention in contradiction with the adjective "fair"). |
23rd February 2011, 07:53 PM | #116 |
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You are the one saying that a particular result (i.e., 100 heads in a row) may not be possible. What everyone else here is saying is that is not correct--all possible unique combinations of 100 random throws is equally possible. Despite all the verbosity, you still have not demonstrated that throwing 100 heads in a row is not possible. |
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23rd February 2011, 07:57 PM | #117 |
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23rd February 2011, 08:00 PM | #118 |
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Did but don't.
In any case, when you ask, effectively, "What can be deduced from a system if I randomly misassign your data about that system?", you're asking a tangential question, to say the least. The issue was never whether a series of 100 values of "T" or "H" was impossible for any system you could imagine. It was whether or not you'd ever actually encounter a run of 100 heads or tails in a fair coin-flipping system, or a human one, on earth. |
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23rd February 2011, 08:02 PM | #119 |
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23rd February 2011, 08:20 PM | #120 |
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Seems to me a fairly easy program to do just that could be written in qbasic, with the randomizer re-randomized via system time for as near random as you can get easily.
Something along the lines of 3 T = 0 7 Y = 0 10 X = rnd(1),0 20 if X = 1 then Y = Y + 1 30 if X = 0 then T = t + 1: print "Only "; Y;" heads.":goto 7 40 if Y - 100 then print "Got 100 heads! Took "; T;" tries." 50 end (Haven't done qBasic in aeons, so I may have made an error there) #EDIT: Couldn't remember how to randomize the timer heh. I could look it up... |
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