probability

by

That blog is our last hope!

Told you, I did.  Reckless am I.  Now, matters are worse:  I’m writing a blog post on a Saturday, because I am going to the office to work today.  I didn’t truly promise, but I did say it was likely.

Speaking of speaking like Yoda (see the opening sentence) I did a little, very brief, voice recording yesterday, as a whimsically silly set of questions arose in my head‒there is nowhere else my questions can arise arise, after all‒regarding an aspect of the Star Wars universe, and I decided to record them.

I didn’t really check my mic placement before I started, so after my quick edit in the form of doing “noise reduction” in Audacity and renormalizing the inherent volume, my voice sounds somewhat weird.  It’s a bit tinny or echoey or something along those lines.  Heck, maybe that’s just what my voice sounds like in real life these days.

That’s pretty unlikely, though.  I’ve heard recordings of my voice, often made by me, since I was quite young (remember those personal cassette recorders in the 70s?).  Still, I couldn’t say with 100% certainty that it isn’t the case.  Indeed, one can never say anything empirical with 100% certainty.

There is after all always the possibility (in principle) of something like Descartes’s imagined malevolent demon, tormenting a mind with entirely illusory experiences.  Anyone who thinks they know some aspect of external reality to 100% certainty is poorly calibrated, doesn’t understand probability, or they’re exaggerating and/or not really thinking about what they’re saying.

Of course, there are many things about which we are so close to 100% implicit certainty that we are willing to risk our lives, usually without even considering that we are taking that risk.  We’re pretty sure of gravity in general, but we implicitly trust the floor beneath us, even in very high buildings.  We’re also pretty sure we won’t die in a car accident on our way to…well, wherever we’re going.  And very nearly 100% of the time, we are correct.

But, of course, every now and then, someone does get killed in a car accident, sometimes on a very short trip, perhaps to the corner store to buy a lottery ticket.  It’s more likely than actually winning that lottery.

They used to say that the vast majority of car accidents happen within five miles of the home.  But don’t worry, once I heard that little bit of trivia, I moved the hell away from that place!

Ha ha.  I have to laugh at my own stupid jokes, otherwise, a lot of the time, no one would laugh at them.

Anyway, as you can probably tell, if you think about it‒though you are not required to do so, your thoughts being your own‒I have no real direction when it comes to this post, and no spontaneously forming topic seems to be appearing, unlike a few times earlier this week.  So, I’m just meandering about in blog post phase space.

That’s okay, though.  It’s Saturday, and I’ve been working all week, and this is my 6th blog post of the week.  WordPress will no doubt send me some automated congratulations on this, my latest “streak”.  They keep doing that, and I know it’s intended to make people feel good about their posting, but it’s just obviously automated and so is annoying.

Also, it sometimes even engages some pathological demand avoidance, arousing a twisted sort of “I’m not doing this for you” feeling that makes one‒well, it makes me‒less enthusiastic about blogging.  The programmed feedback subroutine in WordPress is not my target audience, so getting positive feedback from it doesn’t make me feel that I have accomplished something worthwhile.

Don’t mistake me.  I like getting the specific information, or at least having it available, but it doesn’t have to be accompanied by a cartoon party popper and a “Congratulations!”, as if I’d achieved some kind of merit-based award.  Is this part of the lamentable trend of grade inflation and giving everyone trophies just for participating?

I think some of the mindless, automated, misdirected feedback is part of why I don’t use Brilliant dot org more often*.  They have this “experience point award” thing for when you do problems and exercises and finish sub-courses.  That in itself is okay, because it’s not really too intrusive, and maybe it would be good if you could eventually exchange them for…something, I don’t know.

But instead, they put you in these “leagues” and show you how you compare to other people using the app that day.  That can be kind of annoying, because I don’t go to educational sites to be competitive, except with myself.  I don’t even like multiplayer online games.  And, the trouble is, I get briefly caught up in the league score, because I am intellectually competitive, but that in turn gets distracting and negative (not much, but it’s there) and it discourages me.

I don’t know what I would recommend be done instead; I haven’t really thought about it, I was just expressing a feeling I have about such things.  Maybe other people enjoy these sorts of feedback a lot, in which case, hey, keep it up.  The strength of such enjoyment is almost certainly far greater than my own minor annoyance.

Okay, that’s enough for now.  Below, I am embedding my weird little recording.  I hope you have a good day and a good weekend.

Really, I do hope it, for whatever that’s worth.

*It’s not the only reason nor the most powerful one.  Mainly it’s mental inertia of some kind.

by

The forecast calls for uncertainty

It’s Friday now (as I write this, anyway), and I think that I will have tomorrow off.  But, as some of you may have noticed, the specific plans about my work Saturdays are subject to rather erratic change.  It’s quite annoying; I don’t really like unexpected changes to plan.  I particularly don’t like them when I don’t agree with the reasoning behind them.

Of course, our two most consistently top salespeople at the office contracted when they came aboard not to work on any weekends.  And, as I said, they are consistently our best.  Could there be a causal connection between those facts?  Well, correlation does not necessarily imply causation, of course, but enough correlation should at least shift your credences.

Unfortunately, humans are not naturally good at probability and statistics.  This is part of why I think the subject(s) should be taught in standard education, starting quite early.  Though the subject(s) can be somewhat counterintuitive, the mathematics is not really all that rarefied or difficult, and probability and statistics apply to so much of the world.  On the smallest scales they seem to apply fundamentally.

Anyway, I didn’t come here today to discuss probability and statistics, though obviously I enjoy the subject(s).  So, then, what have I come here today to do or to discuss?  Well, now that I think about it…there is no particular subject.  I don’t know why that should surprise any regular reader, let alone me.

It will probably not surprise you that I have not started playing on Babbel or Brilliant yet.  I do at least look at the apps frequently throughout the day, considering using them and so on.  For whatever that’s worth.

I can allow myself some excuse with Babbel, since it’s difficult to practice a language in a busy office.  But there’s no such reason not to use Brilliant.  Its teaching and exercises are set up in nice, granular ways, so you can do one problem then get called away by work, or whatever, and then go back.

I even don’t mind the rather hokey “experience point” system they use to reward you when you get an answer right.  It’s kind of fun, but it’s not too involved or taken too seriously by the app makers (or so it seems, anyway).  And I definitely have learned new things on the app in the past, and honed and renewed prior skills as well.  So it’s not a waste of time by any means.

The same cannot be as confidently said* about the various apps/sites on which I no longer have accounts.

Of course, time passes‒or whatever it is that time really does‒no matter what we do, and sometimes “wasting” it can be a fulfilling choice.  If we are metaphorical virtual particles then we can behave like them from time to time, not just heading directly to the next interaction, but maybe throwing out an electron-positron pair and then reabsorbing them before they could be detected, or going around the universe and coming at the interaction from backwards in time and behind, as it were, just to show off a bit.

Not everything has to be useful, at least not in too narrow a sense.  Usefulness, like so many things, is in the eye of the beholder.  It is certainly not a universal, general attribute of reality.  So, while it may only rarely be wise to be counterproductive from one’s own point of view, there are times when it’s good‒maybe even useful, ironically‒not to worry about whether something has any point or not.

Yeah, I’m not terribly good at doing that, either.  I don’t know how much of that is due to culture/upbringing and how much of it is genetic or at least neurodevelopmental.  I’d guess it’s not too far from 50/50, but I would not be shocked to find the full truth surprising.

Regarding whether to worry about app usefulness or lack thereof and whether to spend time on the ones that I will have wished I spent time on, well, it’s been said that wisdom, at least a form of it, is the ability to follow your own advice (i.e., the advice you would give to someone else if they were in your circumstances).  I think most people would be able to recognize that, by that particular definition, we are all quite unwise, quite often.

Okay, well, I’ll start to wrap this up.  I really should not be working tomorrow, but if I do, I will almost certainly write a post.  It’s quite unlikely‒I would call it less than 20% likely‒that I will work, but we shall see.  You can check in if you’re “in the neighborhood”.  Don’t look for my posts to be shared on Facebook or Threads anymore, but I do share them on Substack and Bluesky and TWFKAT.  And you can always find them here, directly, and comment if you wish.

Have a good weekend in any case.  That’s an order!

*Well, it can be said, but talk is cheap mother f#cker.  Rather often, people say they are confident and act sure about situations or information that they cannot know with confidence.  I always consider this unwarranted confidence to be a “red flag”, a warning sign that this person’s judgment is unreliable.

by

You’re so vain, you probably think that nothing matters

I was going to start by saying that I had probably written all I could about Friday the 13th and the fact that there are 2 in a row when non-leap year Februaries have Fridays the 13th, and that a first glance might lead one to think this should happen roughly every 7 years on average*.  However, as I noted last time I discussed this, because the leap year day is in February, we will not have the two-in-a-row Fridays the 13th (February and March) as often as we might otherwise; it will not happen every 7 years on average.

Then, this morning, after recalling that today was Friday the 13th, I ran through the next years’ Fridays in my head in the shower, and it occurred to me that the next Friday the 13th in February‒which will be in 6 years, as I noted in the past‒will not be followed by a Friday the 13th in March!  2032 (six years from now) will be a leap year, so there will be 29 days in February, so there will be no Friday the 13th in that March.

The next paired ones, then, will be a further 5 years after that, in 2037 (not a leap year).  It would have been 6 years later, but there are two leap years in that interval, 2032 and 2036, so the next one comes a year sooner than it would otherwise.

It occurred to me that, because of the frequency of leap years, which is almost twice that of the cycles of days of the week, the frequency of those paired dates may well be once every 11 years rather than every 7.  At least those are both prime numbers.  I’m not going to work out some exact formula right now, though.  It’s not really important.

Of course, one could say that nothing is truly important, and I am persuadable along those lines.

There is a Doctor Who Christmas Special (the one from series 5) in which the antagonist/guest protagonist (played by Michael Gambon!) describes a woman in a cryo chamber as “nobody important”, and the Doctor characteristically responds by saying, “Nobody important?  Blimey, that’s amazing.  You know, in 900 years of time and space, I’ve never met anyone who wasn’t important before.”

This is typical Doctor, of course, but it raises the objection Dash (from The Incredibles) voiced when told that everyone is special:  Saying that everyone is important can be the same thing as saying no one is.

Of course, important is in the eye of the beholder.  But then again, the beholder is not important, either, except in its own subjective estimation and perhaps that of a few other, equally unimportant, owners of such eyes.

So, yeah, one could argue relative and subjective importance from local points of view, which is valid but more or less vacuous outside its small scale as far as I can see.  On a cosmic scale, it’s all just dust and shadows.  But you could also say that about the entirety of the cosmos itself.

I guess import has always been subjective, even though people are not inclined to see it that way.  But, of course, people are the products of their “local” forces, and they are not responsible for the laws of nature, nor for the things which have happened in the past that have affected them in the present (which could come under a certain interpretation of “the laws of nature” in and of itself).  I won’t get into all that now.

Going back to the shower, but on an entirely different subject, I was also thinking about the effects of diminishing amounts of shampoo in the bottle on the center of gravity of the bottle.  At the start, when it’s full, the center of gravity is roughly in the geometric center of volume of the whole thing.  But as one uses the shampoo, the center of gravity shifts lower and lower, since the air replacing shampoo in the upper part of the bottle is much less dense than the shampoo or the bottle.

But then, as one gets to the dregs, the smaller and smaller amount of shampoo in the bottle contributes less and less to the overall mass distribution of the bottle and its contents, and the center of mass begins to head back up.  Finally, when the bottle is “empty”, the center of gravity will have returned to almost the same place it was when the bottle was full.

All that’s fairly trivial, well-known stuff, I know.  But it got me to thinking about how much of the laws of physics, such as the laws of gravitation (Newtonian form), are solved using such concepts as the center of mass, which is really just a way of combining and averaging the effects of numerous tiny bits of gravitating material as if they were concentrated at one point.

Much of the mathematics of physics works this way, coarsely approximating the very fine details of reality in a way that provides reliable, reproducible guidelines and can produce testable predictions.

But the granularity of reality doesn’t actually ever go away, not at any level.  Even at the level of the quantum wavefunction of a single “particle”, the actual behavior of the thing as it interacts with things in the “larger” world is the summation of the effects of all the possible quantum states of the electron superposed upon each other and interacting with things‒everything‒which are also just collections of superpositions of quanta.  That superposition happening in a “space” that doesn’t directly coincide with the macroscopic space we experience, but whatever its dimensions are, they are real, because they have durable, reproducible effects.

Mathematics may be unreasonably effective in the physical sciences, as Eugene Wigner famously noted, but it seems not to be a refining of description but rather an averaging out, a glossing over, the inking of an underlying rough pencil drawing which nevertheless still constitutes the real, original picture.

It may be that, in a sense, all science is just various forms of statistical mechanics.  We know that, at larger scales, we definitely need the tools of probability and statistics to navigate as best we can the territory of reality.  And yet, we don’t teach this sort of stuff to most people, ever.  I wrote a post about this on Iterations of Zero, if I remember correctly.

I could go on about all this rather easily, I guess, but I am using my smartphone today, and my thumbs are getting sore.  That’s okay; yesterday’s post was probably way too long, anyway.

If I did a video of my thoughts on this I might be able to get into more detail, though it would probably be even more erratic and tangential than my writing.  Still, maybe it would be worth trying.

In the meantime, I’ll write at you again tomorrow.

*Go ahead, do a search on my blog page for Friday the 13th; I’m all but sure it will bring up the pertinent blog posts.

 

by

This tue, tue sullied (or solid) day

Tuesday or not Tuesday?  That is the question.  And the answer, for today at least, is that today is Tuesday.

I don’t know when you’re reading this, though.  Odds are, if you aren’t reading this on the day I publish it, that you are not reading it on a Tuesday.  In fact, once we get out past the rest of this week* there should only be a roughly 1 in 7 chance that you are reading this on a Tuesday.

That’s probably pretty obvious, huh?  Still, it can be useful to be in the habit of thinking in terms of probability and statistics, since that’s the way nature sorts itself out, right on down to the level of quantum mechanics, to the best of our understanding.  If people had a better understanding of the nature of probability, many things in the world would run far better, or so I suspect.

I have written at least one previous post (on Iterations of Zero) about how I think probability and statistics should be emphasized far more in general math education at the secondary school level (even for non-college-prep students).

Imagine a world in which most people had grown up learning about the nature of probability with reasonable rigor.  There would be fewer headline-based scares about things that are unlikely enough to be irrelevant‒e.g., plane crashes‒and more appropriate understanding about things like vaccination and disease prevention of various kinds, among numerous other matters.

Imagine if the people of the world really understood the difference between absolute risk and relative risk, and if they grasped the difference between sensitivity and specificity for medical tests.  Heck, imagine if the public at large had a decent elementary grasp of Bayesian probability.  Bayes’s Theorem is not really all that difficult, when you get right down to it.  Veritasium did a nice video about it**.

Of course, as I’ve said before, if wishes were horses, we’d all be neck deep in horse shit, whereas that’s only figuratively the case as it is.  But it would be nice if politicians and other people with undue influence had to deal with a general public that was savvy about the legitimate use of statistics and why (and how) they are fundamental to a thorough understanding of the world itself.  It’s not an accident nor a mistake that Jaynes named his book Probability Theory: The Logic of Science.

And science is not an esoteric thing.  It is not a high-falutin’ mode of thought that doesn’t pertain to the average person.  It comes from the Latin scire, meaning to know.  It is fundamental to the nature of our epistemology, to not just what we know about the world but how we come to know it, how complete and how incomplete is our understanding and what the nature of the world really is at deepest and broadest and finest and coarsest levels.

So, it’s fairly pertinent to everyone, really.  After all, if you want to win a game (or get your best score or whatever) you’re best off understanding the rules as well as you can.  A true novice is unlikely to win a game of chess, or of Go, or even of Mario Kart against someone who knows what they are doing.

Now, nature isn’t our adversary per se‒if it were, we would all be long gone‒but it “knows” its rules and always and only plays by those rules, by definition.  In fact, if you come upon a place where you think nature has broken its rules***, what’s really happened is that you’ve come to a place where you don’t understand the rules.  Nature cannot be “wrong”.  There is no such thing as the “supernatural” in reality, because anything that actually happens, that actually exists, is part of nature.

Even if you discovered that you were in a situation such as that described by Descartes or The Matrix, in which the reality you think you know is an illusion, that is simply a newly discovered fact about the nature of reality, and it raises**** the question of what is the nature of that illusion, what is behind it, and by what laws of physics do those entities operate?

So, anyway, it’s good to learn about how reality works if you want your best chance (never a guarantee) of doing what you want successfully and getting what you desire from life.  No one here gets out alive (at least it’s very unlikely) but you might as well make the game as rewarding as you can in the meantime.

*Which I suspect would be when most non-same-day readers would read this.

**He also did a really nice one about the logistic map and chaos and the Mandelbrot Set that will blow your mind if you haven’t thought about it before.

***I’m thinking of those stories with submoronic headlines such as “New discovery breaks physics!” which don’t make sense to anyone who knows anything, and which should embarrass those who write them.

****It does not beg the question.  To beg a question is not to raise the question, but rather to proceed as if it had already been asked and answered in a way that you’re presuming it to be answered.  It is a way of skirting fundamental issues and avoiding having to prove a case.  In other words, it is willfully or accidentally disingenuous.

by

Neither jot nor tittle, but just a title

It is Friday.  Friday it is.  I do not, though, plan to eat any green eggs and ham, nor do I intend to train Jedi.  I merely like to fiddle around with words.  I have also even been known to write and speak about cellos and violins and violas and basses‒wording around with fiddles, that is.

Anyway, this should be the end of the work week for me, so don’t expect a blog post tomorrow.  I’m not saying that there definitely won’t be one; it’s an outcome with a low probability, but it’s not zero.  In principle, the probability of any physically possible event happening is never zero.  But the odds can be so vanishingly small as to be zero for all practical purposes.

For instance, it’s physically possible for the entire Earth (the Moon included) to quantum tunnel to the Andromeda Galaxy, but I wouldn’t hold your breath.  I suspect that the odds of it happening are so low that the time scale between now and the evaporation of the largest black holes due to Hawking radiation (roughly a googol* years) would not even begin to make it likely to happen, even if it weren’t for the fact that the Earth and the Moon will have been so dead and so disintegrated by then that even the memory of their memory’s memories would have been long since lost to any mind that might still exist at that time…probably.

So, you can treat that Earth-Moon Andromeda tunneling as “impossible” for all practical purposes, but in principle, it could happen…

…right…

…NOW!

Okay, well, as far as I can tell, it hasn’t happened.  The sky is too hazy for me to see if the stars have changed, but I don’t think they have.  It would be quite something to experience the local stars of a different galaxy, but of course, if we tunneled into Andromeda, we might be in a relative star desert, or we might be in a place with too many stars for our long-term safety.  Also, if our solar system’s net momentum persisted, we would be unlikely to arrive in any kind of stable orbit of the center of that galaxy.

And, of course, I did not say the sun would come with us‒that would make the whole thing even more vanishingly unlikely‒so we’d all freeze in fairly short order, apart from organisms that use geothermal sources as the base of their food chains and energy cycles.  Those might survive for eons.

Anyway, it’s vastly more likely that I’ll work and write a blog post tomorrow than that we will quantum tunnel to Andromeda**, but it is still a very small likelihood***.  It may be less than one percent, I don’t know.  But it’s quite unlikely.

So, though it might be worth a quick glance to check in come the morning, especially if you were going to do that sort of thing anyway, I would not go out of your way, and I certainly wouldn’t recommend holding your breath.  I don’t think even a sperm whale could hold its breath that long, and I think they have the longest breath-holding record of any mammal (if anyone knows otherwise, please let me know).

In other news‒not that I’ve really given you any news so far‒my keyboard arrived safe and sound (so to speak) yesterday afternoon, so hopefully this morning I’ll be able to finalize the chords to Native Alien.  Then, maybe this weekend, I’ll record a little guitar-chord and voice demo so I don’t lose track of the song.

Then, next week, I can start working on a song based on the trigger “humility”.  I still have no clear conscious notion of an idea for such a song, but I’m not worried about that.  I know I can produce something (not the Beatles song).

I have to keep reminding myself that I don’t need to produce anything great as far as lyrics go‒I think the lyrics I have for Native Alien, which I shared the other day, are okay but not terrific‒I just need to get some words down.  I can always edit and alter things as the process evolves, just as the first draft of a story (or to a lesser degree a blog post) is just the beginning.

I’m also continuing with the circuit course on Brilliant, and I’m alternating reading that book Vector and The Lord of the Rings (yet again) and my own book, The Chasm and the Collision (also yet again, though LotR still holds the 2nd place record for my number of reads, well ahead of CatC and only bested in number of readings by The Chronicles of Thomas Covenant the Unbeliever).

All these are things that I can do alone, of course.  If there’s something to do that would require someone else’s participation, well, I’m shit out of luck.

I think that’s a phrase that applies fairly well to me, come to think of it.  And the word “alone” might as well have my picture next to it in the dictionary.  Though that might be confusing, since I can think of other words that would merit my picture even more than “alone” would‒words that would do their part to explicate just why I am alone, no doubt.

Batman knows I don’t want to hang around with me.

Anyway, I hope you all have a nice weekend, and if anything truly improbable happens to you, I hope it’s a very good improbable thing.

*That’s 10 to the 100th power, or a 1 followed by 100 zeros, in case you’ve forgotten whence the software company cribbed their name.

**Quantum tunneling is not rare on small enough scales, though.  It happens countless times every second in the heart of the sun, for instance.  If it did not, there would not be enough heat and pressure to overcome the coulomb barrier to fusion, and the sun would be some very large equivalent of a brown dwarf…or maybe it would contract more and get hot enough for fusion to take place without tunneling, but then I think the sun would be hotter and brighter and more short-lived, and I think it’s unlikely that the Earth would have produced any life, let alone humans.

***Think about it:  if you took something with odds of ten to the minus 120‒that’s 119 zeroes between the decimal point and the first non-zero digit‒and then made it a billion times more likely than it is, you’d still have odds of 10 to the negative 111th power, or 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001.  This is a good reminder that relative risk (or probability) is not the same as absolute risk (or probability).

by

Is an “almost” pair o’ dice just one die?

Oooooh, it’s Friday the 13th!  It’s so spooky!

Not really, of course.  It’s just a day.  I like Friday the 13ths, mostly just because so many people seem to imagine they are unlucky, though I think that superstition may be less prevalent now that it was in the past.  Nowadays, the day is probably mostly associated with the slasher film “series” that uses that title.  Not that even the original movie’s story ever had much to do with the day.  It just was a catchy, well-known “scary” day, following in the footsteps of Halloween (although the latter at least had a theme that suited the day).

Of course, a major reason I like this day is that the number 13 is a prime number, and I like prime numbers.  I like 13 especially, because 13 is possibly the most feared and reviled of the primes, associated with bad luck in much the way that 7 is associated with good luck.

Hmm.  I know at least part of 7’s appeal probably has to do with the dice game “craps”.  7 is the most common total to achieve when rolling two six-sided dice, because there are more ways to get that total than any other number.  Meanwhile, of course, there is no way to get a 13 on two (normally numbered) six-sided dice, but it is only just out of reach.  It’s the first number that’s too high for such a pair o’ dice*.

Of course, you can’t roll a 1 on two six-sided dice either, but that feels more trivial.

I honestly don’t think the reason for 13’s association with bad luck probably has anything to do with dice; it wouldn’t make too much sense.  But someone out there, please correct me if I’m wrong.

It’s interesting to think about probability regarding dice, not least because the very field of probability theory was first created by a guy who wanted to optimize his chances of winning at dice.  According to what I’ve read, he succeeded, at least temporarily.

Nowadays, of course, that field has grown into a special subset of mathematics and physics and information theory and so on, affecting everything from thermodynamics and statistical mechanics to meteorology and quantum mechanics.  In a certain sense‒given that Schrodinger’s equation describes wave functions that have to be squared (in a complex conjugate way) to get literal probabilities that, based on Bell’s Theorem, cannot be further simplified, as far as we know‒probability may be something truly fundamental to the universe, not merely a tool for situations in which we don’t have access to information.  Based on Bell’s Theorem, which has been shown to apply in the Nobel Prize winning experiments of Aspect et al, it seems that, at root, as far as we can tell, the quantum mechanical operations are fundamentally indeterministic.

Of course, just because something is “random” at a lower level doesn’t imply that, at higher levels of organization, it can’t behave in ways that are very much deterministic in character.  Lots of little things behaving in a locally random manner can combine to create inevitable larger-scale behavior.  Perhaps the most straightforward and compelling such thing is the behavior of gases and the Ideal Gas Law***.  The motion of any given molecule of gas is unpredictable‒at the very least it is stochastic and has so many degrees of freedom as to be unpredictable in practice, but since quantum mechanics is involved in intermolecular collisions, it may truly be random in its specifics.

And yet, when oodles and oodles of molecules of a gas come together****, their collective behavior can be so utterly consistent‒with very little depending on even what kinds of molecules comprise the gas‒as to produce a highly accurate “law” with only 4 variables, one constant, and no exponents!

If that doesn’t seem remarkable to you, either you’re jaded because you’ve known it since secondary school or I haven’t explained it very well (or both, of course).

It’s interesting to think about the probabilities of dice games using more than two dice and/or dice with more or fewer than six sides.  Tabletop role-playing gamers will know that in addition to the 5 “perfect” Platonic solids*****, there are quite a few other symmetrical (but with sides not formed from “regular” polygons) solid shapes that can be turned into everything from ten-sided to thirty-sided dice.

But RPGs tend to involve rolling one die at a time, except when rolling up characters, at which time (in D and D and Gamma World, at least) one uses 3 six-sided dice (or 4 when applying a technique to yield better-than-average characters).

I wonder why there are no games of chance using more than 3 six-sided dice or using, say, multiple four-sided dice or eight- or twenty- or twelve-sided dice.  The probabilities would be more trouble to work out, but they would not be harder in principle.  If any of you out there either know of or want to invent a game of chance using more than 2 dice and/or other than six-sided dice, feel free to share below.

In the meantime, I’ll call this enough for today.  I am supposed to work tomorrow as far as I know, though that’s always subject to change.  If there’s no post here tomorrow, then it probably means I didn’t work.  I probably will work, though I couldn’t give you a rigorous working out of the mathematics involved in determining that particular probability.

Have a good day if you’re able.

*You can sometimes see them by the dashboard lights.

**Unless superdeterminism is correct.  However, this is a very hypothetical thing, and I’m not very familiar with what arguments are proposed to support it, so I won’t get into it.

***PV = nRT if memory serves. [Looks it up]  Yep, that’s right.  Four variables and one constant (R).

****Even if it’s not right now, over me.

*****These are, presumably, solids that really care about each other but in a non-romantic way.