Double Headed Coins and skepticism

[marplots]

Simon's post has me wondering about another question. Suppose you meet someone who tells you they will roll, with a fair die, 5 of the same number in a row. They do so. Now, they remark that this is an amazing thing, but to really convince you of the special nature of this event, they will now roll another two times and get the same number as before each time.

Knowing the probability of this is about 1:46,000 (36 * 1,296); the die is fair, and they made the claim before they rolled -- you are very surprised and wondering about telekinetic abilities, or some unknown cheating method.

However, when you subsequently learn that this person has been doing this same thing an average of 10 times a day for the last 10 years (about 36,000 trials) you realize that at some point, someone would have the experience you had.

My question is whether having the additional knowledge has affected the actual math involved in determining cheating? How can history reach into the present and change the meaning of an event in this way?

 

[Roboramma]

Yet despite that fact, observation of actual monkey-and-keyboard setups demonstrates that they will never produce the string "The quick brown fox jumped over the lazy dog." no matter how long we give them to do it.

If that is true, it's necessarily the case the likelihood of pressing the "x" key after the "o" key (or at least some key after another) is less than it is after a different key.

In other words, if what you are arguing for with regards to coin tossing is true, it's necessarily the case that the likelihood of the coin turning up heads decreases as the number of consecutive heads increases.

Or, to put it more simply, you are arguing that the gambler's fallacy is not a fallacy.

 

[Cuddles]

Each pair is possible, and yet the extended string will never occur, because in reality they have a much narrower range of output than that at that length, with a lot of repetition of identical or similar patterns.

Except that this claim is just as stupid as your claim for tossing coins. You're just assuming some magical change in the laws of physics that suddenly jumps out and grabs the monkey's hand to stop it typing. It really is incredibly simple, so much so that I have to agree it looks like you're just trolling - if something is possible, then it's possible. Yes, it's a tautology. But that's exactly what you're denying. You're claiming that something that is absolutely physically impossible suddenly becomes impossible without any change in the physical setup or the laws of physics. Basically, you believe in God and miracles, you're just not calling it that.

In reality, it's not just possible for the monkeys to type any arbitrary phrase, it's guaranteed to happen, given enough time and enough monkeys. All the observations show is that each key does not have an equal probability of being pressed, therefore it will need more time and monkeys than if you assume equal probability. That's possibly the really sad part - you're spouting these observations of monkeys as if they support your point, but they're absolutely irrelevant to your claims. The monkey observations merely change the probability of key presses and therefore calculations dependent on them. The only difference is a quantitative one, not a qualitative one - it changes from one very small probability to a different small probability. If your god is going to step in and magically make those probabilities 0, it doesn't matter which one you actually start with. You're not just arguing from ignorance of reality, you don't even appear to understand your own argument.

 

[tsig]

It's a fascinating question, and in a way it's one I don't have an answer for, but in another way I do have an answer.

Let's go back to the monkey example for a moment.

There is nothing that prevents a monkey from pressing the "u" key after he presses the "q" key.

And there's nothing that prevents a monkey from pressing the space bar after pressing the "n" key. Or from pressing the dot key after pressing the "g" key.

Yet despite that fact, observation of actual monkey-and-keyboard setups demonstrates that they will never produce the string "The quick brown fox jumped over the lazy dog." no matter how long we give them to do it.
Each pair is possible, and yet the extended string will never occur, because in reality they have a much narrower range of output than that at that length, with a lot of repetition of identical or similar patterns.

Similarly, in the case of the metal cup dropped down the long flight of stairs, it may hit on the same spot on two consecutive steps, but if it's a random drop (and not a carefully orchestrated effort) it will not make identical strikes all the way down, no matter how many times you drop it, because our rough and bumpy world doesn't work like that.

Now, precisely why is that the case? That, I couldn't tell you.

But a truly fair toss of a fair coin, in the real world, when repeated many times in sequence, is exactly the kind of thing that we should expect to reflect the messiness of physical reality. If we find that this messiness is absent -- which would certainly be the case if we got 100 heads or tails in a row -- we would be wise to conclude that the exercise had somehow been shielded from it, or in other words, that it was rigged.

In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum

How do you actually observe a metaphor?

 

[bokonon]

In reality, it's not just possible for the monkeys to type any arbitrary phrase, it's guaranteed to happen, given enough time and enough monkeys. All the observations show is that each key does not have an equal probability of being pressed, therefore it will need more time and monkeys than if you assume equal probability.

I don't believe this is true. Monkeys are far more likely to bang the same key over and over again than to press random keys sequentially, and they'd have to be pressing a variety of keys with frequencies approximating some target language to manage a phrase of even three words.

The situation with monkeys and typewriters is not really comparable to the situation with a fair coin fairly tossed. The constraints are different, and the range of possibilities is vastly different. The results of one have no bearing on the results of the other.

 

[Cuddles]

I don't believe this is true. Monkeys are far more likely to bang the same key over and over again than to press random keys sequentially, and they'd have to be pressing a variety of keys with frequencies approximating some target language to manage a phrase of even three words.

Nonsense. As long as each key has a non-zero probability to be pressed*, any arbitrary sequence will eventually be pressed given enough time. It will certainly take longer if certain keys or patterns are more common, but that was exactly my point - it will only take longer, there is no qualitative change.

* Piggy has explicitly stated this to be the case so I'm assuming it is true for the example.

The situation with monkeys and typewriters is not really comparable to the situation with a fair coin fairly tossed. The constraints are different, and the range of possibilities is vastly different. The results of one have no bearing on the results of the other.

No, the two examples are identical. One deals with a case where two possible outcomes have the same probability, the other deals with a case where multiple outcomes all have different probabilities. That changes the exact numbers in the calculations, but the final conclusion is identical - any arbitrary sequence must eventually occur given enough trials.

 

[bokonon]

Nonsense. As long as each key has a non-zero probability to be pressed*, any arbitrary sequence will eventually be pressed given enough time. It will certainly take longer if certain keys or patterns are more common, but that was exactly my point - it will only take longer, there is no qualitative change.

If, for the sake of argument, we stipulate that a monkey will press every key it touches three times in a row, then the fact that each key has a non-zero probability to be pressed is irrelevant: the word "nonsense" can never appear in the output; the closest approximation would be nnnooonnnssseeennnssseee.

It's the nature of a fair coin fairly tossed that each independent event has a 50% probability. It's the nature of monkey brains and keyboards that repetition will be more common than variation, and adding more time doesn't guarantee "any arbitrary sequence" under those conditions.

 

[Cuddles]

If, for the sake of argument, we stipulate that a monkey will press every key it touches three times in a row, then the fact that each key has a non-zero probability to be pressed is irrelevant: the word "nonsense" can never appear in the output; the closest approximation would be nnnooonnnssseeennnssseee.

It's the nature of a fair coin fairly tossed that each independent event has a 50% probability. It's the nature of monkey brains and keyboards that repetition will be more common than variation, and adding more time doesn't guarantee "any arbitrary sequence" under those conditions.

Fair point, I was being too general there. My apologies. However, the point stands in the specific example since, as noted, Piggy has explicitly admitted that the sequences in question are possible. In a case like his, where the arbitrary sequence is not explicitly disallowed by the conditions, it must come up eventually.

Edit: To clarify a bit further - I am working with the assumption that after a key press, there is a finite, but different, probability for every other key to be pressed. Piggy's statement that the sequences in question are possible supports that assumption. It's possible to set up a situation where that is not the case, such as keys always being pressed three times, but that's not the situation we're talking about.

A related point worth noting is that Piggy claims his argument stems from preferring real world observations over maths. However, the observations he cites could not possibly prove that monkeys never type a certain sequence, all they can show is that they don't tend to do it in within a limited observation time. So he's actually engaging in exactly the same behaviour that he's complaining about in the first place, by taking a limited set of observations and extrapolating to a general case. Even if his extrapolation were correct (which it clearly isn't since it involves magic), it could not prove his ultimate point since in being correct it would prove that such extrapolation can be valid and therefore he's wrong to complain about it.

 

[sol invictus]

If, for the sake of argument, we stipulate that a monkey will press every key it touches three times in a row, then the fact that each key has a non-zero probability to be pressed is irrelevant: the word "nonsense" can never appear in the output; the closest approximation would be nnnooonnnssseeennnssseee.

It's the nature of a fair coin fairly tossed that each independent event has a 50% probability. It's the nature of monkey brains and keyboards that repetition will be more common than variation, and adding more time doesn't guarantee "any arbitrary sequence" under those conditions.

You can clearly set up a sets of rules that - if followed absolutely - will make certain sequences impossible. But what's absurd about Piggie's position is that it requires that there be certain absolute rules that must always be followed (99 heads must always be followed by a tails). That goes sharply against everything we know about science and reality.

In fact, it contradicts Aristotelean logic, as I proved by reductio ad absurdum in post #177.

 

[blutoski]

No, that's not it. It's a long string of physical events occurring in a turbulent environment which reach the same end state. This puts it in a separate category from the types of strings of events -- of many various configurations -- which we would expect, for good reason, to result from actions within such an environment.

Yeah, I'm still seeing the same misunderstanding. Let me rephrase the scenario.

I describe in advance a sequence of 100 flips that are a mix of heads and tails.

The test flips are conducted, and they match my prediction.

Is the predicted sequence more or less likely than the test flips coming up 100 heads.

(my thought: it's the same probability)

The reason we'd be 'impressed' by a psi calling a mixed string in advance is the same reason we're pre-built to be impressed by a string of 100 heads: we've culturally pre-called this one as 'unnatural'.


Something else worth noting: people are really terrible at determining whether a string is random or composed because we think runs are a sign of intelligent input, but mixes are 'natural'. This is why perfect permutations are incredibly rare in long truly random strings, but runs are absolutely the norm.

This is how we caught some psi cheaters - their results were 'too mixed' - they lacked the telltale runs that are typical true randomness. They were obviously cleaned up by a human who thought runs would be suspicious; whereas, they're not really.




Just a comment about the monkeys - the analogy was originally really a joke and not intended to be a serious claim, so it's reasonable to reject it on that grounds.

But it should also be rejected for the following very good reason: monkeys are intelligent creatures with brains that do things with purpose. They are not going to create truly random results in the way a fair coin creates truly random results due to mechanics. And nobody really expects them to.



Magicians take advantage of this. An example is to ask a volunteer to provide a 'random' number between 1 and 10. Most of the time, they will choose 7, sometimes 3, rarely other numbers. Ask for a 'random' vegetable, and people choose carrot almost all the time. A few choose broccoli. If you buy P&T's "How To Play With Your Food" they provide a transparency of St. Carl the Carrot to do a camera trick, because it's that consistent.

The point is: we're really bad at recognizing random vs created. Again, piggy's model is exactly the same as [William Paley's Watchmaker Analogy].

 

[blutoski]

My question is whether having the additional knowledge has affected the actual math involved in determining cheating? How can history reach into the present and change the meaning of an event in this way?

It's not that history is reaching into the present to change the reality of what happened, but that you are able to better evaluate something with more complete information.

It is completely related to the opening post, because this is the impact of prior probability.

It expands to other skeptical topics. For example, healthfraud claims. It's valuable to look at a body of literature rather than just the one most recent study. How does the current study fit into the body of literature?

In your example, it's: How does the current toss fit into the overall toss history for this individual?

A more concrete example is my colleague at work who tells me he knows he's going to win the lottery every damn time he buys a ticket. One day he may eventually win, but he's one of millions who make exactly the same claim and lose week after week. When one of them eventually wins, it's not very impressive because the overwhelming majority of the predictions have been negative. ie: a broken clock is right twice a day, but this doesn't support the claim that it 'sometimes works.'

 

[blutoski]

A large number of highly variable patterns are consistent with our observation of how fair tosses behave at a large scale.

Well, meaning they're the majority of outcomes and therefore more common. That's not saying anything special.

Essentially: you are arbitrarily lumping them together as a 'type' vs another 'type' which is straight runs.

That's a cultural preference - it does not represent a natural phenomenon.

Another poster divided the outcomes into 'looks like Pi' vs 'does not look like Pi' / somebody who reads morse code could interpret Heads as dashes and Tails as dots and have a category 'makes English words' vs 'does not make English words' &c.

But there's no underlying natural phenomenon. It's just playing Bible Code with coin tosses, and that includes straight runs as one outcome that is decreed 'special' because of cultural human meaning.

 

[dudalb]

Every thread on the probablity involved with coin tossing needs at least one reference to this guy:

 

[This is The End]

Every thread on the probablity involved with coin tossing needs at least one reference to this guy:

 

[Malerin]

Gargh. This sounds wrong. I'd like to explain why I think this reasoning is faulty.

Basically, it's inferring cause from results, rather than testing a pre-formed cause hypothesis. Bem just got pwnd for this. (See: [Why Psychologists Must Change the Way They Analyze Their Data: The Case of Psi])

Often times, in statistics, you get a fluky result and have to come up with a cause, ex post facto. I might not suspect a certain dealer is cheating (or have the thought even formed in my mind). However, after the dealer has dealt himself five royal flushes in a row, I would be convinced he's cheating.

This is related to [pareidolia] in that we have a tendency to anthropomorphise 'recognizeable' natural events backwards from the event. ie: we see random numbers, and look for "a pattern" - we're good at finding patterns with completely open criteria.

It's always something you have to watch out for. On the flip side however (no pun intended), that doesn't mean we can never apply a hypothesis to an extremely unlikely event. We do it all the time.

In the world of hypothesis testing, we need to stick to a pre-designated binary criteria, or we've stopped doing science and started doing metaphysics.

When a lottery number sequence comes up a winner, to me it's random. To the person who chose it, it's probably a very special set of numbers. Birthdates or something.

Ahh, now what if the same person wins the same lottery again? We might say it's a fluke. Given how many people play the lottery, there might be some repeat winners. But if they win again? And again? At some point, you become extremely skeptical of chance, and much more open to some rigging of the lottery.

31415926535897932384 is 'obviously cheating' to somebody who recognizes pi, but not to other people. I'm sure mathematicians have a personal inventory of irrational numbers they would recognize but the rest of us wouldn't.

The inability to recogize the significance of a result doesn't mean the result can't disconfirm the null hypothesis of random chance. Let's go back to poker. Suppose my friend and I are watching a poker game. I know how to play, and my friend doesnt't. The dealer deals himself a royal flush. Then another. And another. My friend then asks why everyone has left the game. My friend's inability to recognize the results as meaningful doesn't mean they aren't meaningful. he just couldn't spot it.

Similarly, a Pi result from a bunch of ten sided dice is statistically significant, even though some people won't be able to recognize the significance. Imagine SETI receiving two signals from two systems with possibly habitable planets: one is a series of bursts of static with milli-second pauses between bursts (17489126). Very intriguing. The other is also a series of bursts of static with milli-second pauses: (314159265). Which signal would generate the most excitement?

I think that's what piggy's trying to describe: perhaps a string of identicals is attracting his attention more than other strings because there aren't a lot of special combinations out there he would recognize easily on account of being, well, normal, and not having a math degree or something.

And it's not even restricted to math: the problem generalizes to other disciplines like biology. Behe makes a good living reverse-engineering natural sequences in DNA or shapes of molecules for evidence of intelligent cause.

I think Piggy's point is that while it's theoretically possible for a person to randomnly toss a fair coin heads a billion times in a row, he can confidently say that such a result will never happen. For all intents and purposes, it's impossible.

 

[Piggy]

Ahh, now what if the same person wins the same lottery again? We might say it's a fluke. Given how many people play the lottery, there might be some repeat winners. But if they win again? And again? At some point, you become extremely skeptical of chance, and much more open to some rigging of the lottery.

Or put another way....

When I lived in Florida, there was this guy from Ohio who won the Lotto up there and moved south, but for some reason kept playing. He won a second jackpot in Florida.

Astronomical odds, right?

Well, yeah, but considering how many people play the lottery, it's actually not unusual.

However, if every draw every week for a whole year were to go to a previous lottery jackpot winner, there would be no doubt in anyone's mind that something was seriously wrong.

 

[Piggy]

Could we perhaps agree that it is just very, very, very unlikely to happen any particular time instead of literally, truly, 100% impossible?

I think, with respect to the wishes of the OP, if anyone wants to pursue this topic any further, a new thread should be started.

In fact, this issue has cropped up in several different forms over the years here on JREF.

Can we say that leprechauns do not exist, or must we be content to simply call them "extremely unlikely"? Is "strong atheism" unjustifiable? Is it really true that it's possible that a statue might wave its hand, or that a mixture of gasses might coincidentally segregate? Are we really obliged to concede that Sagan's dragon is merely "unproven" and not "false"?

My stance has consistently been that it is incorrect to hedge on these issues. And I have as yet seen no provable arguments to the contrary.

But this thread is not the place to hash that out.

If anyone's interested, please, let's move it outside.

 

[sol invictus]

The inability to recogize the significance of a result doesn't mean the result can't disconfirm the null hypothesis of random chance. Let's go back to poker. Suppose my friend and I are watching a poker game. I know how to play, and my friend doesnt't. The dealer deals himself a royal flush. Then another. And another. My friend then asks why everyone has left the game. My friend's inability to recognize the results as meaningful doesn't mean they aren't meaningful. he just couldn't spot it.

Again, there is a difference between the assertion "sequence A is more likely to be produced by cheating than it is by chance" and "producing sequence A by chance is impossible".

I think Piggy's point is that while it's theoretically possible for a person to randomnly toss a fair coin heads a billion times in a row, he can confidently say that such a result will never happen. For all intents and purposes, it's impossible.

But (as we keep pointing out) that's quite obviously a false statement. If that particular sequence is impossible because it is very unlikely, then all sequences of the same length are impossible, because they are all equally unlikely - a patently absurd conclusion that can be trivially falsified.

 

[Piggy]

One of the many misunderstandings you evidently suffer from is this idea that long sequences of heads are an "edge" to the space of sequences, or are in any physically relevant way distinguished from any other sequence. If the coin is fair, physics couldn't care less which side humans consider heads and which we consider tails (it's just a label, an arbitrary convention).

Quite correct, physics doesn't know or care which side is heads and which side is tails. But that's entirely irrelevant to my argument.

Physics also does not care which side of an airplane we consider "up" or "down" but this has no relevance to the question of the dynamics of an airplane attempting to navigate a hurricane.

 

[Piggy]

In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum

How do you actually observe a metaphor?

No, the monkey is a real flesh-and-blood monkey.