There has been tremendous confusion here and at Uncommon Descent about what I’ll call the ‘all-heads paradox’.
The paradox, briefly stated:
If you flip an apparently fair coin 500 times and get all heads, you immediately become suspicious. On the other hand, if you flip an apparently fair coin 500 times and get a random-looking sequence, you don’t become suspicious. The probability of getting all heads is identical to the probability of getting that random-looking sequence, so why are you suspicious in one case but not the other?
In this post I explain how I resolve the paradox. Lizzie makes a similar argument in her post Getting from Fisher to Bayes, but there are some differences, so keep reading.
Early in the debate, I introduced a thought experiment involving Social Security numbers (link, link). Imagine you take an apparently fair ten-sided die labeled with the digits 0-9. You roll it nine times and record the number you get, digit by digit. It matches your Social Security number. Are you surprised and suspicious? You bet.
As I put it then:
My point is that for anyone to roll their SSN (or another personally significant 9-digit number) is highly unlikely, because the number of personally significant 9-digit numbers is low for each of us while the space of possible 9-digit numbers is large.
If someone sits down and rolls their SSN, then either:
1. The die was fair, the rolls were random, and they just got lucky. It’s pure coincidence.
2. Something else is going on.
Just as my own SSN is significant to me, 500 heads in a row are significant to almost everybody. So when we express suprise at rolling our SSN or getting 500 heads in a row, what we are really saying is “It’s extremely unlikely that this random number (or head/tail sequence) would just happen to match one of the few that are personally significant to me, instead of being one of the enormous number that have no special significance. Hmmm, maybe there’s something else going on here.”
That helps, but it doesn’t seem to completely resolve the paradox. It still seems arbitrary and subjective to divide the 9-digit numbers into two categories, “significant to me” and “not significant to me”. Why am I suspicious when my own SSN comes up, but not when the SSN of Delbert Stevens of Osceola, Arkansas comes up? His SSN is just as unlikely as mine. Why doesn’t it make me just as suspicious?
In fact, there are millions of different ways to carve up the 9-digit numbers into two sets, one huge and one tiny. Should we always be surprised when we get a number that belongs to a tiny set? No, because every number belongs to some tiny set, properly defined. So when I’m surprised to get my own SSN, it can’t be merely because my SSN belongs to a tiny set. Every number does.
The answer, I think, is this: when we roll Delbert’s SSN, we don’t actually conclude that the die was fair and that the rolls were random. For all we know, we could roll the die again and get Delbert’s number a second time. The outcome might be rigged to always give Delbert’s number.
What we really conclude when we roll Delbert’s number is that we have no way of determining whether the outcome was rigged. In a one-off experiment, there is no way for us to tell the difference between getting Delbert’s (or any other random person’s) SSN by chance versus by design or some other causal mechanism. On the other hand, rolling our own SSN does give us a reason to be suspicious, precisely because our SSN already belongs to the tiny set of “numbers that are meaningful to me”.
In other words, when we roll our own SSN, we rightly think “seems non-random”. When we roll someone else’s SSN, we think (or should think) “no particular reason to think this was non-random, though it might be.”
The same reasoning applies to the original “all-heads paradox”.
So OK, I guess you’re saying that a specification doesn’t really become useful or meaningful unless it is crafted before the event. If I drop a glass on a hard floor and it shatters into thousands of shards scattered about the room, the chance of exactly those shards stopping exactly where they all did seems extremely improbable, PROVIDED I specified them all in advance. Otherwise, it’s a pretty normal result.
So, as ever, the moral of the story is that you can’t determine the likelihood of something after the fact. You have to know at least some of the priors.
Then again, prayers get answered all the time, PROVIDED you are satisfied that whatever happens is the answer to your prayer, however unrelated it may be. Just make the target big enough.
See also: “Texas Sharpshooter Fallacy”.
This doesn’t look to me like a 500 heads situation. It looks like one original flip locked in heads.
As long as you are talking about Social Security numbers, there is another way to look at this so-called “paradox.”
The nine rolls of the die produce a nine digit number. That number has a high probability of matching one of the millions of Social Security numbers that are out there – as well as some that aren’t Social Security numbers.
What is the probability that you would be the person who was watching when the number was yours?
It’s the Lottery Winner Fallacy as seen by the person who won. In other words, all the Social Security numbers that are out there are the “lottery tickets” that have been purchased; and you won.
There was always a high probability that somebody would win.
But since nobody knows who has which “ticket,” or how many tickets were actually purchased, everybody is clueless about the odds.
This is what it means to try to make assertions with no knowledge of the probabilities and the number of “players.” Only people who have such knowledge could know that somebody is likely to win; but even they don’t know who until the die is rolled.
But I return to Sal’s major blunder in ascribing a binomial distribution to the emergence of a particular chirality in the molecules of life. He is asserting a probability distribution without knowing anything about the rules of the lottery, “who” is playing, and how many “tickets were purchased.”
Mike,
It isn’t the Lottery Winner Fallacy. The LWF happens when the probability of somebody winning is 100%, but the probability of any particular individual winning is low. The fallacy occurs because someone argues that “X couldn’t have won the lottery — the probability is too low!” The rejoinder is “Someone had to win, and it was just as likely to be X as anyone else.”
My scenario is quite different. In my scenario, one person is sitting down and rolling one 9-digit number. (Remember, I introduced this thought experiment to show you why it is meaningful to talk about the probability of one-off events.) The person rolling the die “wins” if his or her SSN comes up. Otherwise no one wins.
Since there’s only one person and one number, there doesn’t have to be a “winner”. The odds of “winning” are only one in a billion, and so are the odds of there being a winner. It is truly a remarkable occurrence if you win, just as it is truly remarkable for a person to flip 500 heads in a row. It’s not a fallacy at all to question the result under those circumstances. You’d be crazy not to.
I agree with you about Sal’s blunder. He rejects pure chance and then leaps directly to design, failing to consider the probability of non-design, non-chance hypotheses (which would be very difficult to estimate even if the idea did occur to him.)
But that’s a separate topic.
That’s the point. 500 heads looks fishy and immediately raises our suspicions. A random-looking mix of heads and tails doesn’t.
Yet both of those fixed sequences are equally (un)likely — hence the paradox.
You are close to convert me in an IDst. Because then their argument of probabilities to infer non ramdomnes is true. If the probabilitie of get my SSN makes me infer design, getting a protein, a genetic code or other event with lower probabilities than getting my SSN should make me infer design.
I can’t agree with that. It’s like saying rolling snake-eyes and rolling 7 are equally likely, because each die is equally likely to land on any side. The reason we’re suspicious isn’t because each sequence is equally likely, but because if we consider the number of tails and number of heads in 500 flips, we will find (with a fair coin) that the two will be about equal, regardless of sequence.
Do 500 flips thousands of times and graph the total number of heads from each, and we expect to see a normal bell curve. The further out either tail of the curve we get, the more suspicious we should become.
A sequence 500 heads is suspicious if it indeed reflects a history. Why isn’t it sufficient to say that this doesn’t model biology or reflect any actual history?
There’s the catch, Blas. You’re assuming there was an assignment of significance in advance to a given protein or a given genetic code arising, but that’s not how biology actually works. Any given protein within a given search space may have a low probability of occurrence, but some protein coming up is going to occur. The only ones we really know about are the ones that worked; the ones that failed died off.
I’d like to see SAL or any IDist do this.
–ID ON–
1) Flip a coin until you get 7 heads in a row.
2) Mail it to a friend and give him 3 to 1 odds that he can’t roll a head.
3) He will think you are crazy since the odds should be even on a two-headed coin toss, but he doesn’t know you pre-flipped the coin 7 times and thus the odds on this two sided coin are now 8 to 1 against another head coming up. 🙂
–ID OFF–
As an ID argument, this one is bad.
I think you missed it. The probability of getting SOME SSN is very high. What is the probability of getting SOME protein, or genetic code? Not yours specifically, but ANY?
This is an extremely common ID error, and almost required. Where countless different viable results exist, what you have to look at is the likelihood of getting ANY of them, not one of them in particular. Otherwise, this is what I call the “every bridge hand is a miracle” fallacy.
Don´t push me to be an IDst.
“some protein coming up is going to occur.”
a string of number is going to occur.
“The only ones we really know about are the ones that worked”
We know the keiths SSN come.
Then we have to infer no ramdomness.
keiths,
Probably a dumb question here from a non-stats person, relating to the permutation-combination issue. As has been discussed, even when the experiment involves generating a permutation of H’s and T’s, it’s very natural to consider the experiment where we generate a combination of H’s and T’s (just forget the ordering).
And why not? If our goal was to test whether the coin was fair, then the permutation experiment was obviously the wrong one to perform, so we should have tried something else. In particular, the probability mass function should separate outcomes according to their likelihood given that the coin is fair.
In essence, in converting from permutations to combinations, we have defined an equivalence relation on the permutations (two permutations are “equivalent” if they have the same number of H’s and T’s), and the equivalence classes form the sample space for the new experiment (“carving up the sample space”, in the terms that you used).
The nice thing is that the distribution for this new experiment is not uniform, and the paradox disappears. And it “works”, meaning that it reliably detects biased coins, using elementary hypothesis testing.
So, given that we want to test the coin for fairness, what other equivalence relations could we have placed on the space of 2^500 permutations? Is there an objectively “best” one? (presumably it would have to be permutation -> corresponding combination).
The trouble is, we’re allowing folk to entertain the thought that, during evolution, large functional highly specific proteins (or their genes) were randomly assembled from a soup of amino acids or nucleotides.
Which is most certainly not the case.
Sounds like a variation on the Monty Hall problem. Something that doesn’t respond to intuition.
Exactly!
But we don’t know what protein or gene code is going to come up in advance. That’s the difference you seem to be denying.
A lot of the time, yes, but as you indicate below, the “heads” ratio is a distribution, and so some of the time, the number of heads out of 500 will be unusually high.
This seems to be self-refuting, but perhaps I’m not getting what you mean by “suspicious”. If I sit down and do 10,000 tests of 500 flips of a quarter in my pocket (oh, to have that amount of available free time!), we would indeed expect to see a normal bell curve emerge when charting out “head vs. tails ratio”.
But, in order to make that bell curve, SOME of my tests would have to be 80% heads, or 92% heads, etc. Ratios that are highly unusual — hence the low y-value on the chart! So I complete the test and it’s 94% heads, I plot it way out on the “tail” of my chart.
Am I suspicious that I just did what I did? Do I doubt the quarter flipping I am doing is a ‘fair coin’ flip? The previous 10 trials all happened to cluster right around the 50% heads ratio. Do I trust those previous 10 trials and not the latest, because it’s 94% heads?
You can’t have a bell curve without “tails”. The infrequent but present tests that have unusual balance in the heads/tails ration MUST occur in order for a binomial distribution to obtain.
I’m not pinning “suspicion of the coin flip” on you, here, but there is a problem that keiths is pointing to, that seems to get washed over with “suspicion”, namely that very improbable things don’t happen.
I understand the Bayesian heuristic that attempts some kind of holism in assessing explanations — maybe the coin got switched in a moment when I looked away between tests, and the switched-in quarter is somehow magnetically inclined to heads when landing on my iron table? Etc.
Improbable events do and must occur, and (by defintion), infrequently. There’s a human conceit here, and it’s a very powerful one. If I did sit down and get through 5,000 runs of 500-flips, and THEN I had one test where I observed 80% heads, I’d be less amazed then if I got those same results on my 4th test. But why? If an 80% heads is my “black swan event”, when is it “supposed to” occur?
The occurrence of a black swan event, an extreme outlier, can occur at any point in the trial set. If we did 10,000 sets of 10,000 tests of 500 flips, and graphed the ordinal position of any 80%-heads test, we’d see more of those “black swans” in the middle 50% of our trials than than in the first 5% or last 5%, but ONLY because “50%” has many more chances to catch such an event than the 10% of chances at the ends.
All of which to say: if I were doing my own tests, with a quarter in my pocket, and I got 80% heads on my 12 run of 500-flips, should I doubt my coin, or doubt my eyes? If such a test is statistically inevitable at some point, why am I suspicious of now. If I am at a craps table and watch the shooter roll snake eyes six times in a row, I am plenty suspicious — from a holistic standpoint it’s crazy not to consider designed outcomes and unknown influences on the roll. But sometimes, highly improbable rolls happen, as the must, if and because the dice are fair.
We are animals with an “stance of intentionality”, psychologically. So we are pre-wired to be “conspiracy theorists”. Sometimes there’s no conspiracy and a fluke is just a fluke. I’m continually surprised that this hypothesis gets fully discarded so often. It’s one thing be skeptically suspicious, or even paranoid in fealty to our design-fetished psychology, but dismissing outliers outright as impossible can’t be reconciled with what he know, empirically, about binomial distributions and other probabilistic combinatorics.
This is exactly the point I tried to make way back on Sal’s thread; people are conflating permutations with combinations.
In the case of fair coin flips, every permutation is equally probable; combinations are not. Order matters in a permutation; order does not matter in a combination.
None of my students in my statistics course ever had a problem with this. Granted they were bright students; but the distinction between a permutation and a combination is discussed within the first few pages of most probability and statistics textbooks.
I am surprised I am not seeing those terms used here.
I will wait until folks look up the concepts and start using the terms.
It does look like Sal succeeded by crying squirrel.
He diverted the discussion from chemistry and from the fact that coin tossing is not a relevant model.
Mike,
You’re a bit too quick to assume ignorance or stupidity on the part of your interlocutors.
I’ve been avoiding the word ‘permutation’ for the same reason I replaced ‘substance dualism’ with ‘the soul’ in the title of my other post.
‘Sequence’ or ‘fixed sequence’ can be understood by readers without formal training in probability, just as references to the soul can be understood by people who have never encountered the term ‘substance dualism’.
Have you been following the discussion here and at UD? No one (to my knowledge) is confusing permutations with combinations.
The paradox is far more subtle than that.
Ok, but that is not the problem. We know that are few proteins that are going to work (keith SSN) and it happens to be that that proteins came.
Well; there is no significant difference. It just means that only one person bought a lottery ticket and the probability of his winning is still the same.
But that is really no different. The person who won doesn’t need to know whether or not anyone else had a ticket.
As you may know, not all tickets are sold in any given lottery.
By the way; we are not talking about lotteries in which people get to choose their own number. In those lotteries, several people can win; or nobody wins and the jackpot gets bigger for the next drawing.
As petrushka said, Sal successfully diverted attention from the main blunder by crying squirrel.
Hi socle,
No, there is no objectively best one, and that is what gives the paradox its force.
To test the coin for fairness, you can create a completely arbitrary set of classes (they need not be equivalence classes). You can even assign permutations to classes randomly.
No matter how you create the classes, the test for fairness is always the same. For each class, you simply observe whether the ratio of ‘hits’ to ‘misses’ is about the same as the ratio of ‘permutations in the class’ to ‘permutations outside the class’. Large deviations indicate that something fishy is probably going on.
The paradox comes from the fact that no matter what permutation X you happen to get, there is a set of classes for which X is an outlier — that is, some set of classes for which the class containing X is tiny.
Therefore, the cause for suspicion can never be ‘we got X, but X is a member of a tiny class’. Every X is a member of many tiny classes.
Hence my suggested resolution of the paradox (in terms of the SSN scenario):
Mike,
There’s a huge difference. In the Lottery Winner Fallacy, the error is to think that nobody can win because each person’s odds are too steep. In reality, someone is certain to win (or almost certain, depending on how the lottery works).
In my SSN scenario, by contrast, it’s overwhelmingly unlikely for the person to ‘win’. We are justly astonished (and suspicious) if the person does win.
Do you see your mistake?
keiths,
I am not assuming ignorance or stupidity on your part. However, after nearly 50 years of watching ID/creationists, I am quite familiar with their motives.
I have spot-checked some of the comments here and at UD; I don’t have time to follow every comment.
There are a number of misconceptions I am seeing repeatedly that I don’t normally see in discussions of permutations and combinations. And I don’t mind watching people wrestle with concepts while trying to understand these concepts for the first time.
I am also quite familiar with the history of probability and statistics as well as the more recent pedagogical developments and standardizing of terms. I have been using this stuff for well over 50 years now; it has been a routine part of my working life. Most of us who have been in research deal with it on a daily basis.
Just be sure to separate knowledge of a distribution – whether that is explicit knowledge or implicit knowledge from cultural familiarity – from the observance of a specific event.
You cannot know what the probability of a single event is unless you know whether and how it connects to a larger set of events. And you cannot know a probability distribution unless you already have a specified mathematical model or an empirical set of measurements that encompasses most if not all outcomes of repetitions of an experiment.
These ideas are taught fairly early in science and engineering. It is both explicit and implicit in teaching students how to express an experimental number with the correct number of significant digits plus an uncertainty expressed properly in the correct number of significant digits. You cannot do that with a single measurement.
Sal’s attribution of a binomial distribution to chirality in the molecules of life comes from his imersion in his ID/creationist culture. In that culture, it has become automatic to reify metaphors by assigning the probabilities of simple mathematical models to natural events that they don’t choose to investigate or understand.
Their main purpose is to get attention by provoking debates. It’s a part of their political agenda; they don’t care about the science or proper understanding of mathematical and scientific concepts.
No, there is no objectively best one, and that is what gives the paradox its force.
Thanks, keiths. There’s obviously more to this paradox than I had thought.
Mike,
Of course you can’t infer a probability distribution from a single observation, but that’s not what we are doing in these scenarios.
Instead, we assume a fair coin (or die) and ask “How likely is this observation given our assumption?”
If the observation is “all heads” or “my SSN”, then the answer is “extremely unlikely.” Hence our suspicion. We start doubting our assumption that the coin (or die) was fair.
The paradox is this: Why am I suspicious when I roll my SSN, but not when I roll the SSN of Delbert Stevens of Osceola, Arkansas (or any other random person)? My SSN is no less probable than Delbert’s, after all.
My answer is in the OP.
I think that ID present many other low probability events and as a minimun you are saying abiogenesis is not ramdom.
Yep. All the planets orbit in the same direction. Not binomial. Therefore design.
Of course it isn’t. Have you paid attention to anything on the thread?
keiths,
Then I think “paradox” is not the best word to use here. This is not a “paradox;” it is what happens in the real world.
Your suspicion is because you think you know the distribution (or at least the probability). Your first obligation as an investigator is to repeat the experiment with the same die and with other dice not made by the same manufacturer and to do the experiments under other conditions.
It may very well be the case that the very next experiment with the same die will demonstrate what you expected; namely, not your Social Security number. But whatever the outcome, as a legitimate experimenter, you are obligated to check and recheck and get that probability distribution.
Every experiment in science – especially any really crucial experiment – is checked, not only by the person or group doing the experiment, but by other experiments that will check for systematic errors in the first.
And every experiment has to come with a set of uncertainties attached to the numerical result. The demand for that set of uncertainties means that experimentalists become intimately familiar with their measurement techniques and their instruments. They actually produce those probability distributions.
Experiments are often full of “surprises” and extremely “unlikely” events according to some preconception about the range of outcomes expected. The first thing any experimenter does is a thorough check of the instruments and the constraints and controls being applied to the experiment.
There is always a “shake-down period” in running experiments. During that time one makes sure one is measuring what is intended and that the equipment is functioning properly. In many experiments, one injects representations of confounding data to be sure it is caught and rejected or adjusted for.
Experiments are never one-off events. A single experimental outcome standing alone for several years always creates a lot of activity to try to check it another way. When experiments are difficult and expensive, the tension of “surprise” generates a lot of activity to see if the result makes any sense in a competing theoretical model and many ideas are generated for another experiment that can check the result.
“Surprise” is no surprise in experimental research; it’s expected and planned for. In many important cases, surprises are welcomed because they open up entirely new areas of research.
eigenstate,
Well put.
Sal’s insistence that “the physics of fair coins rules out physics as being the cause of the [500-head] configuration” is bizarre. It isn’t “ruled out” — it’s just unlikely, as is every other specific sequence.
What’s interesting is that Sal would surely not argue that 5 heads in a row are “ruled out by the physics of fair coins”, nor even 10. But 500? Yes, that’s absolutely “ruled out”.
It’s as if he thinks that when the probabilities get small enough, they actually become zero. Amusingly, it’s the flip side of Joe G’s belief regarding large numbers:
Replace ‘number’ with ‘probability’, ‘large’ with ‘small’, and ‘infinity’ with ‘zero’, and you’ve pretty much got Sal’s misconception.
Nnnnggghhh. Still failing to see the relevance of this to chirality. One way to end up with a string of heads is to start with one (p = a mere 0.5, well withing the UPB) and breed from it, with lateral duplication. Another way is to find oneself possessed of a coin-grabbing template whose contours match only one enantiomer. And breed from that.
That’s my reading of the history. One first coin toss followed by replication.
We could envision several scenarios:
1. Some bias in chemistry that we don’t fully understand.
2. A very rare and possibly unique series of events leading to the first replicator.
3. Several OOL events followed by a competition that eliminated one side.
4. Something else.
But once you have a successful replicator, there are no more possibilities of changes in handedness.
Mike,
Here’s what you’re missing. Suppose I am unable to repeat the experiment. For whatever reason, I get to roll one — and only one — 9-digit number.
If I roll my own SSN, I will be suspicious, even though it is a one-off experiment that I cannot repeat — and rightly so. My suspicion is justified.
On the other hand, if I roll an SSN that, as it turns out, belongs to Delbert Stevens of Osceola, Arkansas, then I will not be suspicious.
The paradox is: Why am I suspicious when I roll my SSN, but not when I roll Delbert’s? They are equally unlikely.
We would always prefer to have more experimental data, of course. No one (except maybe Sal) would claim that a single observation is enough to settle the question definitively.
None of that changes the fact I can be justifiably suspicious on the basis of a single observation.
Being suspicious does not absolve you from having a testable hypothesis. You stipulate that you are unable to test the dice for fairness, but this is unrealistic. Why not?
Second, there are six or seven billion people in the world and only a billion 9 digit numbers. I’m willing to believe that someone somewhere has rolled their SS number, even if they didn’t notice.
Aside from the minor detail that dice really don’t produce the range from 0 to 9.
Allan,
There isn’t any, except in Sal’s mind. That’s why I call it the ‘all-heads paradox’ and not the ‘chirality paradox’.
As is so often the case, we’ve ended up discussing the mistakes people make on their way to the original issue, rather than discussing the original issue itself.
Not to worry. Several of us have pointed out that even if Sal could rule out ‘homochirality by pure chance’, he hasn’t established ‘homochirality by design’. We’re ready to reiterate that if, after licking his wounds, he tries to advance the argument again.
eigenstate,
One of the better antidotes to falling into the habit of always seeing conspiracies is to stay engaged with the real world. The fortunate part of our having evolved in a real world is that it provides feedback.
Pattern recognition in observing the real world is what we have evolved to do.
However, pattern recognition of events that take place only within our own heads can pretty quickly mislead without feedback.
That is one of the reasons that the experimental sciences can be so invigorating to those who like science; the universe seems friendlier as one gets use to the “surprises.” It is far richer than expected.
Petrushka,
I have a testable hypothesis: that the die is fair and the rolling process is random.
As I wrote to Mike:
A single observation is enough to cause me — justifiably — to doubt my hypothesis that the die is fair and the rolling process is random.
Sure. No one (except maybe Sal, again) is arguing that it is impossible. I’m just saying that if I sit down and roll my SSN, then suspicion is warranted, even though I have only one observation on which to base my suspicion.
I covered that in the OP:
You said you could not repeat the experiment, which means that no hypothesis can be tested.
Your new hypothesis is technically a superstition. It is not a hypothesis unless it can be tested.
Like many people I can suspect conspiracies when certain kind of events happen. When people die in strange car accidents after saying they have evidence of wrongdoing in high places, my spidey sense tingles.
But I do not have evidence, nor do I have any way of testing my conjectures. So they remain just another odd thing in life.
Science doesn’t really work with odd, one-off events that leave no evidence.
keiths,
Well, I don’t see the world that way.
If I am not allowed to check it, then either I am living in a harsh dictatorship of some sort or I no longer have access to what I need to repeat the experiment.
The result of a one-off experiment is simply the result of a one-off experiment; I don’t know anything from a one-off experiment.
The outcome may be a surprise, and I may never be able to check it. Unless I have some other reason to believe the results, they remain unverified and inconclusive.
In the real would, I would be finding ways to check the result. I realize that there are many people who believe anecdotal accounts of a single event and believe them without question. That is not how scientists are trained to think; nor is it in the psychological makeup of most experimentalists I know.
That attitude is what comes from years in the lab. The world responds to testing. Without follow-up experiments and interlocking results, nothing is settled.
“Suspicion,” “paradox,” “surprise;” I don’t want to get into word games.
Petrushka,
The hypothesis is in place before I roll the die. Rolling the die is in fact a test of the hypothesis.
The paradox is that some specific outcomes make me suspicious, and others don’t, even though all specific outcomes are equally probable. That seems wrong, but it’s not, for the reasons I put forth in my OP.
Mike,
Sure you do. A single observation of a black swan can falsify the hypothesis that all swans are white.
It also works with probabilistic examples. My provisional assumption is that the die is fair and the outcome random. A single experiment — rolling the die nine times and seeing my SSN come up — leads me to doubt my hypothesis.
I regard my hypothesis as less likely after the experiment than I did before. It’s a classic Bayesian inference. A single observation gives me useful information.
People who play their favorite numbers at Lotto have a testable hypothesis, and with great regularity, one or more of them wins.
To activate suspicion, the odds need to be a bit higher than one in a billion.
keiths,
The paradox is this: Why am I suspicious when I roll my SSN, but not when I roll the SSN of Delbert Stevens of Osceola, Arkansas (or any other random person)? My SSN is no less probable than Delbert’s, after all.
Ok, just had to reread the OP; for some reason I latched onto the wrong end.
Sal’s insistence that “the physics of fair coins rules out physics as being the cause of the [500-head] configuration” is bizarre. It isn’t “ruled out” — it’s just unlikely, as is every other specific sequence.
Wow, that is bizarre. In an experiment with a uniform distribution, some outcomes cannot occur (absent shenanigans), while others can, presumably? Surely he must know better. 😯
Mike,
I think he’s alluding to the fact that HIS SSN is pre-specified, whereas Delbert’s was only determined after the fact – gee, we got this number, I wonder if it’s anyone’s SSN? Hitting any pre-specified 9-digit number is a surprise. You don’t need more than one shot to know if you hit the target, if you knew what the target was before shooting (and were not permitted to either aim or look).
But beyond this, figuring out the mechanism behind your reasonable doubt might be a challenge. You’d need a hypothesis proposing HOW a sequence of die throws produces your SSN. Nothing so simple as loaded dice, for sure!
The probability that such a mechanism even exists, might be less than the probability that it just happened by chance.
Petrushka,
When I run the SSN experiment, I am really comparing the probability of two mutually exclusive hypotheses:
1) the die and the rolling process are fair; vs
2) the die and the rolling process are not fair.
Before the experiment, I have no reason to doubt hypothesis #1. The die is symmetric, it doesn’t seem unbalanced, and I’m unaware of any way in which the outcome could be rigged.
Now I sit down and roll it nine times. Out comes my Social Security number.
Am I just as confident in hypothesis #1 as I was before the experiment? No way.
After observing that result, any sensible person would lower the probability of hypothesis #1 and raise the probability of hypothesis #2.
Does that mean that the outcome was rigged? No, it still could have happened by chance. The probability of hypothesis #1 is lowered, but it doesn’t become zero (that’s Sal’s mistake).
A single observation has enabled me to rationally and justifiably alter my assessment of the two competing hypotheses.
Even if you say in advance you are going to roll your own SSN, it’s not really a hypothesis unless you have a conjecture regarding causation.
Are you specifying that the same sequence will always come up?
Are you asserting you control what comes up by psychic powers?
Are you specifying someone has the dice rigged by remote control?
Does it add anything to the discussion that Las Vegas uses such patterns to spot cheaters and card counters?
They seem to be using ID filters, but they have hypotheses regarding causation.