Over at The Skeptical Zone Elizabeth Liddle has weighed in on the “coins on the table” issue I raised in this post.
Readers will remember the simple question I asked:
If you came across a table on which was set 500 coins (no tossing involved) and all 500 coins displayed the “heads” side of the coin, how on earth would you test “chance” as a hypothesis to explain this particular configuration of coins on a table?
Dr. Liddle’s answer:
Chance is not an explanation, and therefore cannot be rejected, or supported, as a hypothesis.
Staggering. Gobsmacking. Astounding. Superlatives fail me.
Not only is Dr. Liddle’s statement false, it is the exact opposite of the truth. Indeed, pharmaceutical companies, to name just one example, have spent countless billions of dollars in clinical trials of drugs attempting to rule out the “chance explanation.”
Don’t take my word for it. Here is a paper called What is a P-value? by Ronald A. Thisted, PhD, a statistics professor in the Departments of Statistics and Health Studies at the University of Chicago. The abstract states:
Results favoring one treatment over another in a randomized clinical trial can be explained only if the favored treatment really is superior or the apparent advantage enjoyed by the treatment is due solely to the working of chance. Since chance produces very small advantages often but large differences rarely, the larger the effect seen in the trial the less plausible chance assignment alone can be as an explanation. If the chance explanation can be ruled out, then the differences seen in the study must be due to the effectiveness of the treatment being studied. The p-value measures consistency between the results actually obtained in the trial and the “pure chance” explanation for those results. A p-value of 0.002 favoring group A arises very infrequently when the only differences between groups A and C are due to chance. More precisely, chance alone would produce such a result only twice in every thousand studies. Consequently, we conclude that the advantage of A over B is (quite probably) real rather than spurious.
(emphasis added)
In a clinical trial the null hypothesis is that the apparent advantage of the treatment is due to chance. The whole point of the trial is to see if the company can rule out the chance explanation, i.e. to rule out the null hypothesis that the results were due to chance, i.e., the chance hypothesis. So, if “chance is not an explanation” what is the point of spending all those billions trying to rule it out?
Want more? Here’s a paper from Penn State on the Chi-square test. An excerpt:
Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. For example, if, according to Mendel’s laws, you expected 10 of 20 offspring from a cross to be male and the actual observed number was 8 males, then you might want to know about the “goodness to fit” between the observed and expected. Were the deviations (differences between observed and expected) the result of chance, or were they due to other factors. How much deviation can occur before you, the investigator, must conclude that something other than chance is at work, causing the observed to differ from the expected. The chi-square test is always testing what scientists call the null hypothesis, which states that there is no significant difference between the expected and observed result
(emphasis added)
Obviously, asking the question, “were the deviations the result of chance, or were they due to other factors” makes no sense if as Liddle says, “chance is not an explanation.”
I don’t know why Dr. Liddle would write something so obviously false. I am certain she knows better. “Darwinist Derangment Syndrome” or just sloppy drafting? I will let the readers decide.
First off, I assume that by Darwinist you mean scientist? Darwins work only applied to living systems, not the origin of life. All the scientists I know use the phrase ‘we don’t know’ to say ‘we don’t know’ but I guess what you mean is that any event that is not the result of some agents intention is ‘chance’? – or as you seem to be putting if – If God didn’t do it then it was chance.
No, I mean some scientists and some not scientists.
I hope you do not use it saying the ususal “we do not know how it evolve(but we are sure it did) but no doubt we are going to figure it out”.
No, from a scientific point of view the universe should be determined or ramdom with or without God. With God ( or with any other non material beens) there is the possibility to be ramdom or determined with the ramdomness or determination possible affected by God.
Then my point is
Blas,
I agree that there are scientists who take evolutionary theory beyond the physical realm to make metaphysical claims, and that it can be very annoying and frustrating. Having said that, the problem is not with the theory itself but their conflation of science and philosophy. And that’s a problem ID has unwittingly adopted by accepting the false premise that random events are unintentional events. That too is a metaphysical claim that extends beyond the limits of science, and in the opinion of a good many Christians – and scientists – is in error.
So humor me for a moment and consider the possibility: the randomness of evolution was intended and its outcome was foreknown. Would you still have an axe to grind?
Right — and “chance” means quite different things in science and in metaphysics.
Let me check through RB’s post:
Absolutely correct. The null hypothesis in Thisted’s paper is that there is no difference in effect between placebo and treatment. It is not “chance”.
Exactly. The null hypothesis is that the means in the population to which the inference is being generalised do not differ. However, samples may, because the samples are random – in this case, placebo and treatment are randomly allocated to two groups assumed also to be randomly drawn from the population (although this is unlikely) to whom the conclusions will be generalised. This is where “chance” in.
Exactly. The “causes” here are assumed to be actual factors affecting the outcome of the disease being treated, including genetics, diet, general fitness, age, etc, which are too complex and too interactive to model separately. As a result there is unmodelled variance which accounts for differences between individuals and thus between groups. Chance is still not “causal” here.
Absolutely correct.
Absolutely correct.
Absolutely correct.
Absolutely correct. Text book answer. Thisted would undoubtedly agree.
Quite.
I’d just love to know in why in any particular Barry thinks that RB’s assertions are wrong. I can’t see a single one that is.
We use the phrase ‘we don’t know’ to say ‘we don’t know’ – this means that we don’t know how or why something happened.
I’m not sure what you mean by ‘the usual ..’ but perhaps you are referring to current gaps in knowledge?
If God is interfering with something then it is not random, nor is it determined in the sense of being deterministic.
Genetic drift is the change in the frequency of a gene variant in a population due to random sampling.
Are you having problems with the concept of random sampling, or are you trying to claim that if it is stochastic then it is ‘unknown’ …?
I’m having a hard time understanding your point here? Perhaps the term random sampling is to generic. How about this:
Genetic drift is the change in the frequency of a gene variant in a population due to events that are unrelated to the fitness advantage or disadvantage that the gene variant confers?
I haven’t been following the whole argument, but has Barry Arrington had a change of heart since August:
“So when you say: “When the Bayesian approach tries to adjudicate between chance and design hypotheses…” that makes no more sense,[B] because chance is not an explanation.[/B] Just as a Fisherian approach does not attempt to determine whether data are unlikely to be “due to chance”, but rather to determine whether, under some null hypothesis, a random sample would resemble our data sample. So the Bayesian approach does not try to adjudicate between any hypothesis and “chance”. What it does is to adjudicate between two proposed explanations for your sample of data.”
OH NOES! LAW OF NON CONTRADICTION! IS THERE ANY DEPTH THESE IDISTS WONT SINK TO? SLIMEY SLIME SLIME. Etc.
um, those are my words, Richard, in a post addressed to Dembski!
Do’H! Those long single-color comboxes on Uncommondescent made it look like a run-on of Barry’s! Feel free to tidy the thread, if you wish.
Yes, thanks for pointing out the first sentence. The ‘universal (or generalised) Darwinism’ or ‘selectionism’ of Campbell, Dawkins, Heylighen, Cziko, et al. are obvious examples. One problem is there is not a single theory; there are multiple evolutionary theories in multiple fields or disciplines.
Another major problems is that many scientists, trained as they are in a narrow naturalistic speciality, are almost illiterate in philosophy. The converse is of course also true, that most philosophers are almost illiterate in natural (or social) sciences. The thinkers who do cross-over work offer opportunities for specialists to widen their awareness of context and think outside of narrow research boxes. That’s why recent work involving ‘philosophy in science’ is also rather promising.
Another problem is the growth of interdisciplinary thinking is still catching on and will take many years to challenge and begin to repair the fragmentation of knowledge that has taken place since the rise of ‘modern science.’ This interdisciplinary work will erode the confidence some natural scientists have in ‘naturalism’ by exposing them to non-naturalist approaches (which of course doesn’t necessarily mean ‘super-‘ or ‘supra-naturalism’). This means that the study of ideology is not just a negative or pejorative political move, but rather essential for better understanding the biases, distortions and blind spots sometimes held by specialists narrowly trained.
The larger problem is with naturalism, not Darwinism per se, as Darwinism is only one variety of naturalist ideology. IDists are too narrowly focussing on Darwin and missing a broader conversation. And the constant appeals to probabilism and specificationism in IDism have already become boring.
By ‘Darwinist,’ Blas said: “I mean some scientists and some no[n-]scientists [sic].”
So he seems to invite philosophy or theology/worldview into his argument, even while IDT obscures this. What else could he mean by invoking ‘non-scientists’?
In the end, rhampton, my guess is that Blas (along with many of his skeptic opponents here) doesn’t like your approach because it appears to open the door to science, philosophy and theology/worldview collaborative discourse. And because IDT is *supposed* to be a ‘strictly natural scientific’ theory, i.e. because Dembski and Behe, Meyer and others strangely view ‘Darwinism’ as a strictly natural scientific theory that they aim to overcome with their ‘new science’ involving Intelligent Causes, it won’t help Blas’ promotion of IDT to tell what his axe to grind is all about. It is simple for IDists to simply claim people who reject it do so because they misunderstand it. What is hard to admit for IDists is when people reject IDT who actually already accept lowercase ‘intelligent design,’ i.e. that the universe is created by a Mind, yet who have seen through IDT’s facade as currently shown by IDM leaders.
IDists, almost exclusively theists (in contra-distinction to naturalists), are too busy enjoying their USAmerican culture war dance with ‘brights’ to face this possibility without spinning it:
Lizzie:
I’m pretty sure Barry knows that RB is right. He can’t admit it, however, because that would make him look like a… well, like an Arrington.
rhampton, to Blas:
That hypothesis only makes sense if the Intender intended to make it appear that there is no Intender.
Why is he/she/it hiding?
There may be a bright side to all this.
After the UD crowd reifies their thinking and irons out their differences about probability, perhaps we will see one of them come up with their definitive version of an ID/creationist “second law of thermodynamics” from the “law of large numbers” that will put the final nail in the coffin of “Darwinism.” They seem to feel they are almost there. 😉
I’m not holding my breath in giddy anticipation, however.
Those assertions are are wrong in the same way that ‘Darwinism’ is wrong.
They just are wrong, OK?! They know that even if they can’t articulate it, just take their word for it and ID will take it’s proper place in the grand scheme of things.
It’s a gut-feeling kind of thing, say “you are wrong” often enough and perhaps the jury will start to believe. Except, Barry, the jury in this case does not seem to be swayed by your just wrong style of “argumentation”.
He’s not hiding, it just that you’re not looking.
God’s a carpenter in jeans and you are looking for a physicist in a lab coat.
eh, nothin’ new. its always been that way.
…if god is so powerful, how come he cant ….blah, blah, blah.
…..on the other hand, if we are so smart, how’s come we can’t see jack shit.
…any day now, it’ll come to us…that missing element that turns leaden amino acids into golden organisms…..but it aint’ god….we know that….who’s ever seen god!!! Sheesh!
Tell it to that child that just died of starvation. Sorry kid, you just were not looking hard enough….
“the Intender intended to make it appear that there is no Intender.”
Admit that ‘possibility’ (as your proper capitalisation indicates) and the landscape of discussion changes; universal naturalism (cf. anti-theism, without an Intender) cannot then be considered the only option.
“Why is he/she/it hiding?”
Have you never given an anonymous gift, keiths? Since you capitalised ‘Intender,’ then likely it would make sense to capitalise He/She/It too. And that identity would be ever beyond limited human capacity to fully explain ‘why’ in the ‘scientific’ way you seem to be asking it.
Why does God use chance ‘in nature’? Don’t ask IDist-theists (or their preferred atheist dancing partners, several of whom are to be found at TSZ) to willingly speak to this question. They want polemics, a culture war between ‘design’ and ‘chance.’
Genetic drift is the change in the frequency of a gene variant in a population due to random sampling.
Are you having problems with the concept of random sampling, or are you trying to claim that if it is stochastic then it is ‘unknown’ …?
I’m having a hard time understanding your point here? Perhaps the term random sampling is to generic. How about this:
Genetic drift is the change in the frequency of a gene variant in a population due to events that are unrelated to the fitness advantage or disadvantage that the gene variant confers?
Neutral mutation A get fixed by “drift”.
Neutral mutation B do not get fixed by “drift”
WHich is the process and the parameters that makes mutation A get fixed and not B?
That `s the point. Metaphysical conclusion based on “science”.
That is the other side of the error scientific conclusion based on metaphysic.
“That is the other side of the error scientific conclusion based on metaphysic.”
Please try again. What you wrote is not coherent in English.
What’s wrong with natural science and metaphysics working together?
rhampton’s point is sharper than Blas seems to realise.
I don’t understand. Are you after a specific example?
I think the general answer lies in the mechanics of reproduction and the numerous factors that determine which genes from each parent get inherited. You might be better off asking a biologist to explain it because there are an awful lot of variables involved – but I think the net effect is random sampling which is why it is referred to as random sampling.
Then you are deterministic given the neutral mutations A and B always A will be fixed and B no.
No. I have no idea how you could come to that conclusion based on what I have written.
Like I said, ask a biologist. What I do know is that the entire reproductive process is stochastic in the sense that any result of genetic replication is not precisely predictable (You cannot predict the exact genetic code resulting from replication). But that does not preclude statistical predictions being made about the overall makeup of the resulting genome. ‘Chance’ does not capture the complexity of the event in any useful way. Random sampling is better but it is still a description of the outcome of a process.
What you seem to be saying is that unless you believe that randomness does not happen (i.e. you believe in determinism) then any outcome that is the result of a stochastic process was caused by chance.
Is this correct? If it is then consider the fact that biological development is also stochastic. Would you consider yourself to have been cause by chance simply because your development from embryo to adulthood included some apparently random elements?
—
Consider these statements:
Event A was caused by chance
Event A happened by accident
Event A was caused by accident
Event A appears to be random
Event A was not predictable
Can we use accident and chance interchangeably (some people do), and if we do, how can accident be a cause.
Interesting discussion, I am just going to jump right in here.
It seems to me, in a test of tossing coins, if the null hypothesis is, “is this a fair test of coin tossing?” how is that any different from stating that the hypothesis is that this is a random coin toss (iow, a chance event.)
I can agree that in more complex hypotheses the null hypothesis is confirming whether or not what you are testing is true, but in the case of a coin toss, you are only testing one thing: “is this chance or not.” I feel Dr. Lizzle has really complicated the matter by pretending that the question of “Is it a fair coin toss hypothesis” is somehow different from the “Is it a chance coin toss.” Or even one could say “Is it NOT a chance coin toss.” They all mean the same thing. Take your pick, call it a chance or not chance coin toss, what’s the difference.
Then you simply do your little test, and use the data to confirm the null hypothesis or not. Thus the null hypothesis is chance.
I think her word games have confused the issue way too unnecessarily, and unjustifiably.
I contend that this is the same hypothesis.
You’re exhibiting the same type of sloppy-thinking behavior as they do over at UD. Just like them, you fail to read for comprehension and in context. You misrepresent Lizzie’s clarification of key scientific terms as merely “word games”, using exactly the same pejorative phrases they do.
What’s your nym over at UD? If you’re not one of them, you should be embarrassed that your shallow comment can so easily be mistaken for one from their playbook.
hotshoe,
Your hand waving and sloppy word commemorating aside, the fact remains. An hypothesis of “Is it a fair coin toss?” is the equivalent of “Is it not a chance coin toss.” You didn’t address this.
If this is the quality of your retort, I probably won’t respond much to you in the future, other than to perhaps dismiss you.
phoodoo:
Speaking of “sloppy word commemorating” (sic).
The explicit null hypothesis is “a fair coin, fairly tossed” with the corresponding probability distribution, which includes each toss being independent. As Lizzie has demonstrated, there are many other possible distributions, none of which corresponds to the generic abstraction “chance”.
I suggest you read her post Chance and 500 coins: a challenge again to help you understand what a null hypothesis is and why “chance” doesn’t qualify.
Thanks, Patrick 🙂
Patrick,
Patrick, I am well aware of what Lizzle is claiming. She is claiming that a fair coin fairly tossed is a different meaning from “a random coin toss randomly conducted.” She admits that we have to call it random, even though in its truest sense it is not exactly random, because there are physics involved which determine the result.
Yet an hypothesis of “Is a fair coin fairly tossed?” holds the same meaning as “Is it a random (chance) coin toss.” One would not be wrong to use this as their starting hypothesis. Dr. Lizzle doesn’t get to decide the wording of the hypothesis she prefers, just because it sounds better to her.
No, it doesn’t, for reasons Patrick and Lizzie have already pointed out.
Tell me phoodoo, why do casinos routinely win about 5% on roulette when “it is a random (chance) wheel spin”?
thorton,
Tough one.
The psychology of gambling addiction?
Gin and Tonics?
Because they have two green spaces?
But the players and the house are both playing the same game of “chance”. Why don’t they win equally then? According to you it’s both just “chance”, right?
How about another example. Let’s play poker. You have to play the five cards you’re dealt, I get to discard and redraw up to three times. Should be a fair game since both our final hands are from “chance”, right?
No, it doesn’t, no matter how much you hop up and down repeating “is so” “is so” “is so”. We already know there’s going to be a physical result caused by forces we won’t make the effort to (or can’t) measure — that’s the chance part but that’s the boring part. The only interesting part is the not-chance question: is this really is a fair coin, and is it really is fairly tossed? Well, we don’t start out assuming that our gambling friend is cheating, so our null hypothesis is that the coin and the flipping process is fair — not that it’s “random/chance”; nobody cares about that — and our job as rational thinkers is to figure out a way to test the null hypothesis that the coin/flips are all fair.
No, you would not only be wrong, but also stupidly wasting your time.
Umm, yeah, she does get to, because words actually do mean things, and scientific hypotheses actually do need to be worded clearly and precisely to be testable, and people like the UD idiots don’t get to cheat by equivocating words in an attempt to make their sloppy thinking look “like science”.
If you want to participate in a real science forum like this one, instead of in a cargo-cult “science” forum like UD, you’re going to have to step up your game.
Ha, is this called stepping up your game?
What does fair mean in this context? When is it a fair coin toss and not a fair one?
Don’t you mean when the coin toss is random (chance) and when it is not? Well, yes of course you do. At least people who can think do. Not people who just proclaim they are scientists because they make stupid cliches like, “step up your game.”
You are quite right. It takes more to make someone a scientist then using clichés.
For example, an actual scientist might have addressed some of the points made to them rather then just ignoring them like they do not exist. As you have actually done in the last half dozen or so posts alone.
So perhaps when you start to act like a scientist you can tell other people how they should be acting themselves. Until then, not so much…
OMagain,
What does “fair” mean? Light skinned? Equal rights for all? Or does it mean random?
Fair in “a fair coin fairly tossed” means an unbiased coin and an unbiased toss with a probability of heads on each toss being exactly 0.5. When you perform many such tosses you get the balanced probability distribution curve Lizzie showed in her first diagram here.
Random merely means the results are stochastic and can have a probability of heads other than 0.5. Trials where the probability is not exactly 0.5 give skewed distribution curves as shown in Lizzie’s other examples.
Random does not mean fair. Precise definitions are important in science which is why IDers avoid proper definitions like the plague.
Hammer, nail, head, Thornton.
“Random” does not mean “fair”. They aren’t even related, according to the thesaurus.
[Merriam-Webster]
Funny, no “fair” in those synonyms for “random”.
Now let’s go to “fair” and see what its synonyms are. As already noted, “fair” has multiple meanings, so the definitions/synonyms have been grouped together understandably [by Merriam-Webster].
Hmm, funny, no “random” listed anywhere in that whole page as a synonym for “fair”.
hotshoe,
Using a dictionary is apparently not your strong suit, Mr. Hammer:
unbiased, unbiassed [ʌnˈbaɪəst]
adj
2. (Mathematics & Measurements / Statistics) Statistics
a. (of a sample) not affected by any extraneous factors, conflated variables, or selectivity which influence its distribution; random
http://www.thefreedictionary.com/unbiased
Yea funny isn’t it. I guess they don’t teach that skill in skeptical school.
So you’re yet another IDer only interested in playing rhetorical games and not interested in understanding the concepts. Got it.
You never told me if your think my proposed poker game with you is fair because it is random. Well?
protip. Wall of text generally will get you minus points. Exactly the same CSI could have been transferred by simply saying the word is not found in this definition.
Oh wait now, I think I’ve just advanced ID. I must therefore be an ID scientist.
When do they start sending the cheques? Do I have to actually write a book?
thorton,
You couldn’t even get that the reason the house always wins in roulette is because of the two green spaces which increases their odds, and you want me to answer you about a silly poker game? And now when you have been called out about your brilliant deduction that fair means “non-biased” but it doesn’t mean random, when I just showed you in mathematics that’s exactly what it means, you want to call it rhetorical games?
Do you have the intellectual honesty to ever admit when you are wrong? Does hotshoe? (Those are rhetorical questions, btw-look it up in the dictionary if you need to.)
More evasion of the question and silly rhetorical games.
Fair processes have outcomes that are random.
Not all random processes have outcomes that are fair.
Only a specific type of random process, one with a uniform probability distribution, is considered fair by the definition used in statistics.
Random still does not mean fair no matter how much you equivocate.
Phoodoo,
Lot of mistakes here, I will try to deal with the major ones.
1) “Fair” and “random” are NOT synonyms, as Thorton pointed out to you.
(Let me also join in in encouraging you to read Lizzie’s 500 coins thread. Chance and 500 coins: a challenge)
“a fair coin fairly tossed” means that p(Heads) for each toss is 0.5, and that the tosses are independent. One way of ensuring independence would be to draw a sample of coins in an unbiased manner from an infinitely large population of coins that are 50% heads. Drawing the coins at random would be unbiased, thus “unbiased” can be used when describing a sample from a population as meaning “random”. But Thorton was referring to an estimator of probability, not a sample, so the appropriate definition of “unbiased” would be
which you omitted from your dictionary quote-mine.
2) [This one is a whopper] you said:
Say what? “use the data to confirm the null”.
NOOOOO! You “fail to reject the null”. This does not mean you “confirm the null”.
I will try to illustrate, in case you are still confused after reading the thread linked above.
I have a couple of strings of 500 1’s and 0’s, and I wish to test the null that these are a representation of “fair coins fairly tossed”. I test the total number of 1’s against the expected distribution under my null. In both cases, the total is within the expected distribution under my null. So I have FAILED TO REJECT my null. It is NOT confirmed. In fact, the first string is a representation of a short stretch of ASCII. Doing some further tests on this first string, I am able to reject the null that these “tosses” are independent: there is way too much auto-correlation, and the distribution of runs of 1’s and 0’s is clearly wrong.
However, the second string passes all my tests: I FAIL TO REJECT the null for this second string. This still does not mean that it is the result of “fair coin fairly tossed”.
It is an encrypted message.
DNA_Jock,
I stated that the hypothesis could either be asked in the affirmative or the negative. If the hypothesis is that it is NOT a fair coin fairly tossed, then actually you would be confirming the null, not failing to reject the null.
So if we are looking at 500 coins which appear to be spilled on a table, all showing heads, we can use this SAMPLE to decide that clearly this sample was NOT unbiased, and thus it was not a random sampling of tossed coins (Ah ha, someone has clearly been toying with these coins!). That is how the analogy applies to design. I did not play any rhetorical games. The only possibilities in our test are heads or tails, all 50% chances, and so we are looking at chance or no chance. Biased or not biased. Fair or not fair, they are the same meaning in this case.
One can always make a concept more confusing if that is their goal. Lizzle used the words fair, but that doesn’t mean those are the words that one has to use, despite some people’s holy allegiance here.
phoodoo,
Not all random processes give rise to a uniform distribution. Throw two fair dice and count the total number of points. A seven will appear six times as often as a two or twelve.
Your error of logic has already been dealt with. There are other non-intelligently-guided processes (i.e. ‘random’) which could have resulted in 500 heads on the table. The best of many offered were that there were originally 1000+ coins spilled on the table, roughly half heads. The tails down side design gripped the table surface firmly but the heads down was quite slick. A local train that passed every hour vibrated the table and cause the ‘heads down’ coins to ‘walk’ and fall off the table. The net result is 500 coins, all heads, left on the table through completely random processes.
As Dr. Nick Matzke pointed out to the UD ignoramuses over a week ago, the probability assessment you come up with depends on your assumptions about the history of the test. That’s why ‘fair coins tossed fairly’ is a reasonable null but the impossibly vague ‘chance’ is not.
How long until phoodoo grasps this simple concept?
Interesting. Given the hypothesis “that it is NOT a fair coin fairly tossed”, what is the predicted probability distribution against which you will compare your data? You don’t have one, because your hypothesis lacks specificity. Your hypothesis is not a NULL hypothesis.
Wrong again. It could be a perfectly unbiased SAMPLE, if the POPULATION is (almost) entirely heads-up. The words ‘random’, ‘unbiased’, ‘fair’ (generally meaning equiprobable) and ‘independent’ have specific meanings that matter.
“Chance”, not so much.
Have you read the “Chance and 500 coins: a challenge” thread yet? It is a great demonstration of different stochastic processes that yield quite distinguishable results, clearly showing that Barry’s “The Chance Hypothesis” is so vague as to be quite useless.
So very true. [sigh]