For even more edification, read this article in German: Denken Sie an eine Zahl zwischen 1 und 10.
I’m picking a number between 1-10. What is the probability of you guessing it correctly?
It doesn’t exist. No probability exists.
“C’mon, Briggs. It’s one outta ten. Don’t play games.”
Well, the whole setup is a game. But ignoring that, no. It’s not “one outta ten”. It’s not anything.
“What BS. It’s ten percent. Anybody can see that. What kind of statistician are you, anyway?”
Well, I’m the Statistician to the Stars!, so I have that going for me. Still, I insist. There is no probability.
“Do I really have to do this? Look, you coulda picked 1, you coulda picked 2, and so on up to 10. That’s 10 choices. That makes one outta ten.”
Nope. Tell you what. Have a go a guessing and I’ll prove that you’re wrong.
“Funny man. Okay, I read most people pick 7, so I’ll say 7.”
Nope.
“Fine: 6.”
Nuh uh.
“This is getting tedious. 1? 2? 3? 4? 5?”
Keep going.
“8? 9? 10?”
None of those.
“Bull! It has to be. You’re reminding me why I stopped reading you.”
My number was 6.3.
“What!? Hold on. You didn’t say anything about fractions. Always cheating.”
You’re misinformed. I’ll repeat what I said: I’m picking a number between 1-10. Last I checked. 6.3 is between 1-10. You didn’t get it, and you got the probability wrong, too.
“I see what you did there. You’re trying to say that all probability is conditional, and if you don’t specify the conditions, you can’t have a probability.”
That’s right.
“And part of those conditions was the meaning of the words ‘I’m picking a number between 1-10.'”
Yep.
“I assumed an integer as one of my premises, and you defined it as any number. Meaning the words and grammar of any probability problem matter.”
They always do.
“Funny man. Hang on. You said 6.3, so I assume you could have picked any number, any real number. Right?”
Right.
“But there’s an infinite number of those numbers! If that’s what you meant, then there’s no probability of you picking. How could you even pick if you had to first select from an uncountable number of numbers?”
Good question. I can’t, not if I’m presented with an uncountable number from which I have to select one. I don’t even know how to define “picking one” when the number of numbers is so huge I don’t even know how to comprehend them all, except by the fiction of pointing to a symbol.
“Ha! Hoist meet petard.”
I blush.
“Wait, though. You did pick one.”
I did.
“Meaning you had to have some mechanism for picking. Meaning you couldn’t have been picking from some hugely impossible uncountable number, but from some smaller set.”
That follows.
“There’s no way I could have known what that set was, or the mechanism of picking was.”
True.
“Best I could do was to suppose your behavior was similar to other people’s. I could’ve used that as a premise, and then deduced a probability from that.”
That’s it, all right. Which is what you did when you guessed 7.
“But you, you have to be different. Funny man. So you have to pick a non-integer, just to be a funny man.”
My jokes are world famous.
“Yeah, sure. But there was no way I could’ve guessed exactly what you were going to do. And even if I did, I’d have no way of knowing how many digits you were going to throw out.”
Very true.
“Which means there was no way I could really deduce a quantitative probability—not unless I accepted premises which were too concrete.”
That’s because not all probability is quantifiable. That’s what the man said in this award-eligible book. That not everything has a number is a hard equation to swallow for some, growing up as they do, devoted to scientism. Still, it’s true. The only proof there is that everything has a number is hope.
As in so many things, Thomas Berger has the best thing to say about our obsession with numbers. Here’s Jack Crabb, returning to the company of whites after having lived with plains Indians for many years (start of Chapter 8).
That’s the kind of thing you find out when you go back to civilization: what date it is and time of day, how many mile from Fort Leavenworth and how much the sutlers is getting for tobacco there, how many beers Flanagan drunk and how many times Hoffmann did it with a harlot. Numbers, numbers, I had forgot how important they was.
As important as they are, they are not the most important things. Quality triumphs over quantity.
“You’re reminding me why I stopped reading you.” Self-awareness!!! 🙂
Now I know who wrote the section of the Troll Manual™ that explains how to nitpick language!
(The Troll Manual™ is the notes used by trolls on blogs to try and tear apart everything they don’t like. It consists of numerous trite answers and is repeated ad nausem in the hopes of winning through intimidation. Nitpicking is one chapter.)
Naive Question: Doesn’t your argument also call in to question the notion of uncountability?
It reads like you are walking the fine line between something having no probability and something having zero probability.
In your example, the guesser should have a chance at guessing the right number. Calculating the probability of getting it right, given the premises offered, should tell us that there are an infinite (uhgg) number of possible answers, each just as valid as the others. That means that the guesser could never be certain of his guess; the probability of it being any given number is zero.
This should be distinct from something having no probability. A vacuum has no matter in it but is is not nothing. It is a vacuum. The game you play above has a probability associated with each guess. It is zero. It’s not nothing.
Or did I skip a chapter?
Being a security professional, my standard answer to any question is, “Well, that depends.”
Because the answer to almost any question depends upon the circumstances surrounding the problem, the availability of solutions, imagination, convention, the budget, political considerations, and the vagaries of the solution approval process and the personalities associated therewith.
There is a finite but incalculable chance that I’m happy you have progressed this issue.
Douglas,
No, not the math, which is fine. But our understanding, which is not. Ordinary counting infinity is already nigh impossible to think about in terms of uncertainty of a thing that can actually take infinite states. Then comes uncountable. As long as the path to infinity, and the separation between things in actuality and things in potentia are made clear, then we can do it. Otherwise we risk fooling ourselves. See Chapter 10 (I think) of Jaynes.
Art,
A proposition having zero probability would mean it is false. Just like in logic, where you interested in the truth or falsity of a proposition but can’t demonstrate it without some kind or argument (which premises includes knowledge of the words and grammar and meaning of the symbols used to write it), you cannot have a probability of a proposition with an argument.
Aww, crap. I used my sophisticated Monte Carlo model programmed in SAS, R, and Python, and it guessed 6.29999999999999. I think I forgot about floating points or something…
Fairly random introduction of petard to hoist, where both seem to be nouns. “Hoist” is typically a verb when used with “petard”, being the past tense of “hoise”.
VG
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Loved this as am new to it all! Am forwarding it!!
God bless, C-Marie