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18th February 2011, 06:54 PM | #1 |
Illuminator
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Impossible coin sequences?
Is a sequence of 100 heads in a row literally impossible to get without cheating?
This was discussed in another thread but a mutual decision has been made to start a new thread about it. http://www.internationalskeptics.com...d.php?t=200394 My position is that it is entirely possible to get that sequence without cheating. Each flip is approximately 50/50, regardless of what came before. All heads is as likely as any other single sequence. |
18th February 2011, 06:58 PM | #2 |
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18th February 2011, 07:17 PM | #3 |
Metasyntactic Variable
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Agreed.
To believe otherwise is to fall for the Gambler's Fallacy; "... the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future."WP |
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18th February 2011, 07:40 PM | #4 |
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Given that you have, for example, 10 coin tosses, and the results are either H or T, the sequence HHHHHHHHHH is no more unlikely than any other, such as HTHHTHHTTH. It is us humans who attribute certain results a special meaning, even though statistically they're no less likely to happen. So yes, it's perfectly possible to get a sequence of a hundred heads. It's just as possible as it is to get any other result, in fact.
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18th February 2011, 07:50 PM | #5 |
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18th February 2011, 07:50 PM | #6 |
Extrapolate!
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The first quote in post #2 really clinches the matter in an intuitive and obvious fashion: all sequences are possible because we can relabel them in an arbitrarily predetermined way, and there is no reason for the universe to care about our labels. I don't understand Piggy's reasons for rejecting this.
--- Throughout that thread, I couldn't make sense of Piggy's argument. Perhaps it's because I live in mathland, but at times he seems to invade that land as well: This is suggestive of some sort of "space of sequences", with the special sequences of runs "live" on the edge. So presumably they're special because they're... hard to reach, being on or near the edge? But for any sequence, there's exactly one possible way to obtain it: the coin has to sequentially go through the exact prescription of the sequence. So they're all "on the edge". If one insists on thinking of the space in terms of reachability, then it looks like a tree: Code:
* * = flip T/ \H * * <- 1-sequences: (T),(H) / \ / \ * * * * <- 2-sequences: (TT),(TH),(HT),(HH) etc. |
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18th February 2011, 07:59 PM | #7 |
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It's also entirely possible to bowl a 300 game, though very much less likely than any other bowling score.
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18th February 2011, 07:59 PM | #8 |
Illuminator
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It is a very clever argument but I'd say it's a bit difficult to understand.
I love the part that states that if one of them is impossible, then all of them are impossible. |
18th February 2011, 08:00 PM | #9 |
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which would be 1:2^100, but only before you start tossing the coins, and it's the same for *any* sequence of a 100 coin flips. And clearly it's not impossible to get a sequence of 100 coin flips.
where people get confused is not understanding that the probability of throwing a head, after having thrown 99 heads in a row, is 1:2 |
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18th February 2011, 08:03 PM | #10 |
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18th February 2011, 08:08 PM | #11 |
Penultimate Amazing
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Yes, bowling a 300 game is much less likely than any other score, because it's not random. It involves skill. With tossing a coin 100 times, getting 100 heads in a row is no less likely than getting another similarly unique sequence, even if it looks like a random mix of heads and tails. |
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18th February 2011, 09:12 PM | #12 |
The Clarity Is Devastating
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Here's another way to clarify the issue:
As explained already, any specific sequence of 100 flips is equally possible, so of course they're all possible. What is not possible is to predict the sequence that will occur, with any practically significant degree of confidence. So, it's absolutely not possible to confidently predict HHHHHHHHHHHH.... Nor, of course, HTTTHTTHHHTH... or any other specific sequence. But the only absolute impossibility is of confidently predicting. That impossibility does not rub off onto the possibility of any specific sequence occurring. And thus, it does not preclude the possibility of the sequence you predict occurring; or in other words, the possibility of correctly predicting. Those are merely astronomically unlikely. Respectfully, Myriad |
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18th February 2011, 09:37 PM | #13 |
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It's discussion like these that sometimes make me question the sanity of some of the most brilliant posters here.
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18th February 2011, 09:48 PM | #14 |
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What is your position?
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18th February 2011, 10:00 PM | #15 |
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Thanks, Alan, for starting the thread.
It's nearly midnight here, way past my bedtime, and I have appointments in the big city tomorrow, so I have to do a drive-by (sorry, Sol) but I now have this thread on my subscription list. First I'll say that I have no objections to the stats regarding the range of possible outcomes of a system which produces all mathematically possible configurations of T/H for a given series length. As far as the gambler's fallacy, just to give a kind of preamble to my way of thinking, let's rephrase it this way.... Suppose you're betting on a run of 100,000 fair coin tosses that has already occurred. At random, the result of 50,000 of those tosses are revealed. As it turns out, 75% of the revealed results are heads, and only 25% are tails. What odds are you willing to take that the remaining 50,000 will be evenly split among heads and tails? Would you accept even money on a bet that 25,000 (+/- 2,500) of the unrevealed flips ended up heads? Or would you be inclined to bet that the majority of the unrevealed flips were tails? In other words, would you bet that the one-off event (the revealing of a random selection of results) was the abberation, or would you bet that the extended series of events resulted in a significant abberation? |
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18th February 2011, 10:03 PM | #16 |
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18th February 2011, 10:22 PM | #17 |
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I'll elaborate more later, but my position is that the statement you're responding to is inherently flawed.
The real question we need to be dealing with is this: Given a supposedly fair toss, is a completely smooth results-space more or less likely to indicate cheating than is a non-smooth results-space (regardless of the exact configuration of the non-smooth space)? By analogy, let us consider the task of determining whether we are in the ocean or a swimming pool by means of examining the behavior of the water within a nearby radius. If we detect low turbulence and no tidal effect, then we are either in a swimming pool, or in an unusually stable sector of ocean. Suppose that the circumstances do not change over the course of 2 hours. Assuming that we know we are somewhere in the real world, do we conclude that we might be in the middle of an unprecedented anomaly, or that we're in a swimming pool? If we listen to the logic of the post you're responding to, we simply count the "extremly unlikely" low-turbulence/no-tide scenario as one of a large number of equally unlikely possible configurations. But if we use our faculties of reason, we will realize that there are a large number of configurations that conform to "we are in the ocean" and a much smaller number of configurations that conform to "we're in a pool". So this business of claiming that a run of 100 heads is equally un/likely as any particular scenario of mixed tails and heads is a red herring. The important feature to recognize is that a vast number of mixed heads/tails configurations will be typical of the results-space of a long series of fair coin tosses, while a long series of only heads or tails is typical only of a rigged system. The appeal to the supposed equal likelihood of any one particular configuration is a misapplication of statistics to the actual situation on the ground. |
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18th February 2011, 10:24 PM | #18 |
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18th February 2011, 10:27 PM | #19 |
Extrapolate!
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It's not a particularly relevant question, because as in the other thread, multiple people have tried to make you see the explicit distinction between
(1) whether for a fair coin, all sequences are equiprobable, and that a particular high-run sequence is just as likely as any other particular sequence when you're predicting before the fact, and (2) whether in observing a coin that gives a very high run of heads, it is reasonable to conclude that the coin is not fair (which is actually simpler than the situation you've just posited). You seem to be rhetorically arguing that the answer to (2) is a resounding 'yes'. I don't think anyone has said otherwise. Moreover, that answer is perfectly consistent with answering 'yes' to (1). For a fair coin, no high-run sequence is any way more special than any other sequence. What can be considered special is the event of getting some (nonspecified) high-run sequence, i.e., the set of high-run sequences is much less numerous than the set of sequences without high runs. You clearly recognize this: And then Sol followed up in #123: No one disputes your claim about concluding unfairness, at least in that particular case. The dispute is entirely based on you not admitting that (1) is a completely correct statement in its own right that's also consistent with (2). |
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18th February 2011, 10:33 PM | #20 |
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The problem with your ocean vs pool analogy is that I could be in a lake. Is that a pool or the ocean. I could be standing in a puddle and fit every other criteria. The other problem is that you're trying to use an analogy when one isn't really needed.
If I flip a fair coin, and the first two results are HH, what likelihood is the third flip to be H? It's the same chance that it will come up to be T. |
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19th February 2011, 12:19 AM | #21 |
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19th February 2011, 01:41 AM | #22 |
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As has been stated many times already, any specific sequence of 100 flips is just as likely as any other.
But if you were aiming to come up with a specific sequence you'd have to target something like 100 straight H's just because of the difficulty of manipulating a genuine coin the way you want to. You'd choose a double-headed coin and go for the 100 H's. Has a sequence of 100 H's ever happened? In the other thread sol said: "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)." Consider every coin flip that's ever happened (whether observed or not, such as the way a given coin lands when you put coins on a table or into a machine). Put a figure on the average number of coins in existence per day, the number of flips experienced per day and the number of days humans have been using coins : Average number of coins : 10^10 Average flips per coin per day : 10^2 Total days coins used : 10^6 These are probably wildly inaccurate numbers, but it doesn't really matter. Clearly we've had nowhere near enough trials to even remotely expect to find a 100H sequence has actually happened. In fact I'd guess the Earth isn't going to survive long enough to see it, but breakfast calls too loud for me to bother with rough calculations. Piggy's original stance seemd to be that if you saw a 100-flip trial under way and beginning with 15 H's then you're watching a biased trial. I would agree to the extent that I'd love, at that point, to get some money on the sequence continuing to 100 H's, assuming the odds were attractive. Evens would be plenty attractive enough for me |
19th February 2011, 01:57 AM | #23 |
Illuminator
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That is not the topic of this thread. What Piggy was arguing was that it is literally, truly impossible to get 100 heads in a row.
Originally Posted by GlennB
We have been seeing sequences with an equal chance of happening every time there are 100 coin flips, and sequences less likely to occur every time there are 101 coin flips.
Quote:
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19th February 2011, 01:58 AM | #24 |
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And I feel like I have to clarify because it is embarrassing me that some people responding might have misunderstood: I am not advocating for the point of view that it is impossible.
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19th February 2011, 02:34 AM | #25 |
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19th February 2011, 02:37 AM | #26 |
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I literally do not understand Piggy's stance at all. Not one bit.
He must realize that saying one specific outcome of 100 flips is impossible is the exact same thing as saying that each and every other possible outcome of 100 flips is impossible. |
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19th February 2011, 02:42 AM | #27 |
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The chance of getting one head in a row is one half, two heads a quarter and so on.
The chance of getting 100 heads in a row is one in 2^100 (that's 2 x 2 x 2 ... with 100 twos) That's a big number - it's approximately 126765 followed by twenty five zeros. The number of atoms in a coin is only about 5 followed by 22 zeros. Now lets assume that every time you flipped a coin you caused one atom to be worn away from it. Let's also assume that once 20% of the coin has worn away, you can no longer tell heads from tails, so you have to swap to a new coin. If the coins were nickles, you'd (on average) have to wear out about six million dollars worth for each run of 100 heads to occur. Now you can already see that it's practically impossible to get a run of 100 heads. And, of course, the assumption about only wearing one atom away per flip was ludicrous. No matter how careful you are, you would cause much more wear than that so you'd actually need a whole lot more than $6 million, and a correspondingly longer amount of time spent flipping |
19th February 2011, 03:08 AM | #28 |
Illuminator
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Does anybody who is saying it is unlikely think that they are arguing against me?
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19th February 2011, 03:10 AM | #29 |
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On average, yes. However, since the sequence in question is just as likely to occur at the beginning of the experiment as in the middle or at the end, it is not practically impossible, only (as I believe everyone but Piggy has been saying) as unlikely as any other 100-trial sequence.
Every time you flip a fair coin 100 times, something truly unique and practically miraculous just happened. You'll never see another run quite like that again. Makes me shiver just to think of it... |
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19th February 2011, 03:12 AM | #30 |
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I prefer to start with 2^100 coins, flip them all then see aside all the tails.
Repeat with the remaining coins 99 times. |
19th February 2011, 03:18 AM | #31 |
Illuminator
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I am now volunteering to conduct that experiment. I just need people to donate the coins.
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19th February 2011, 03:23 AM | #32 |
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19th February 2011, 03:25 AM | #33 |
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If you compare the likelihood of 100 heads versus anything other than 100 heads, then yes it is extremely unlikely.
If you compare the likelihood of 100 heads to any other single combination, it is equally as likely/unlikely. And renders the selection of any combination of heads or tails (yes, even if 'all' are heads) an arbitrary one. Assuming I'm wrong here; what makes the selection of 100 heads so special that it's mathematical likelihood is reduced? I could arbitrarily pick any single combination, and very likely not see that combination reproduced ever again. |
19th February 2011, 03:26 AM | #34 |
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19th February 2011, 03:28 AM | #35 |
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From what I've read, Piggy is distinguishing between everyday life (it is impossible) and probability-theory-world (it is possible). Elsewhere I've noticed people calculating that it would take many universe ages of coin-flip trials to reach the point where H(100) could be expected to be witnessed in a certain time frame. I believe this is Piggy's point.
Different folks are using the word "impossible" in different ways. |
19th February 2011, 03:38 AM | #36 |
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Piggy's point is quite clear. The fallacy, is saying that there's something special about H(100).
And with that; where do you draw the line at impossible? H(90)? H(70)? H(40)? H(20)? H(10)? My understanding, is that putting a limit anywhere is simply incorrect, and meaningless. |
19th February 2011, 03:50 AM | #37 |
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19th February 2011, 03:58 AM | #38 |
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19th February 2011, 04:11 AM | #39 |
Illuminator
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I sometimes call things impossible in 'everyday' conversations. But this is not that. It was a discussion about the extremes of probability theory! If somebody doesn't take the extremes of probability theory into account when talking about just that very topic then it's like somebody not taking chemicals into account in a discussion about chemistry.
Even if you are correct and the possible/impossible question was a misunderstanding, Piggy wrote some very strange statistical claims about its likelihood that we could discuss anyway. |
19th February 2011, 04:18 AM | #40 |
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