Story Probability

Yesterday I was re-watching Arachnids in the UK, the fourth episode of the latest season of Doctor Who, and a somewhat goofy idea popped into my head about how to respond to the charge that sometimes stories are just ‘too improbable’ to enjoy — or to have happened at all.

That certainly is an accusation that seems to apply in many cases. In order for some story to have happened at all, certain events had to happen just so and in the right order. It’s easy to shake your head and think, “Yeah, right. As if that could actually ever happen.”

For many years I’ve had a generic response to that accusation, but yesterday I realized it can be justified mathematically!

Now don’t be scared; I’m not going to get into any heavy math lifting, just some general descriptions — no actual math required, and no numbers were harmed in the making of this post.

(Also, all of its electrons are free-range, mostly solar, and 98.3% recyclable.)

In fact, if it’s not obvious, this is all a bit tongue-in-cheek.

(Yet it makes as much sense as anything else these days, so let’s continue.)


Let me get the mathy part out of the way. It’s just one thing: how weird things get if we take infinity seriously.

There is reason to not take infinity seriously. After all, nothing in the physical world is infinite.

At least not in the ‘there’s always another’ sense. A circle, in some sense, is infinite, it goes around and around forever. But there’s no such physical thing as an infinite supply of circles.

Yet circles are, in that sense, infinite, and there is always another number after any given number, so numbers are genuinely infinite. Infinity has some reality, it seems.

(It even turns out that numbers come in at least two “flavors” of infinite: countable and uncountable. And mathematicians hold that there are an infinite number of flavors for infinity!)

The tension between nothing physical being infinite and math itself having infinite infinities might be taken as evidence to the artificiality of math — that it’s something we made up, a game of symbols.

But that is a long-running philosophical discussion I only mention to set the stage. For now we’ll accept infinity (the countable kind at the least) as a real thing.

As a real idea, anyway (like justice or unicorns).


Accepting infinity leads to some odd things (such as the Hilbert Hotel). In this case, the weirdness has to do with probabilities.

Imagine for a moment that we have a magic bin with an infinite number of objects in it. Imagine also that one-trillion of those objects are red; all the rest are blue.

If we reach into the bin and remove an object, what are the odds that we get a red object?

It turns out the odds are indistinguishable from zero. There is effectively no chance whatsoever that we will get a red object, even though there are one-trillion of them.

This is because the odds in question, infinity-to-one-trillion, are impossibly one-sided. To see this, let’s divide both sides by one-trillion.

Which gives us odds of infinity-to-one.

We could divide by even larger numbers, but we’d only reduce the right side of those odds. The left side remains forever infinity.

Keep dividing, and it effectively becomes infinity-to-zero.


This holds true even if the right side is (countably) infinite so long as the left side is an uncountable infinity.

For example, the natural numbers are (countably) infinite, while the real numbers are uncountable. (Given any real number, we can’t even identify the next real number, which is what makes counting them impossible.)

Therefore, given a bin of all numbers, the odds are effectively zero that we would draw a natural number rather than a real one, despite the infinite supply of natural numbers.

(In cosmology, this is known as the measure problem.)


Okay, enough math; the point is simply this:

Given an infinite supply of something, the odds on drawing a member of any particular sub-group of that supply is effectively zero.


Now, let’s imagine the infinite supply is stories.

All the stories.

Most of them — very nearly all — are ordinary, even boring, tales where nothing exciting or surprising happens. Certainly no weird coincidences, paranormal manifestations, alien appearances, or costumed superheroes.

But if all the stories are there, then so are the interesting ones.

So are the ones involving massive coincidences and momentous events.

So are the ones based on that single moment when life changes.


But those are the stories worth telling and hearing.

Whether parable or entertainment, those are the stories we crave.

In part, no doubt, because they represent such rare exotic fruit.

Perhaps also because they represent ideals, goals, dreams, hopes.

Stories are a distillation of our experience, concentrated, purified, curated, even aged (compare the original Grimm’s Fairy Tales to the Disney versions, for instance).

If real life is an apple; stories are fine apple cider (or jack).


So of course they’re improbable, some of them.

My old response was along the lines that stories didn’t need to be about what usually happened, or what ought to happen logically, but what actually did happen that one time.

My new response can be along the lines of Math! and isn’t it amazing the author managed to find such an interesting gem among all the ordinary stories!

And isn’t it great they have that drive and make the effort, because most of that infinite supply is pretty dull. (Like the one where a guy spends an hour writing a blog post. Or the one just now where someone spends a few minutes reading one.)

(Props to storytellers and perhaps an insight to why I never thought of myself as one: no regard for my own skill at finding interesting gems.)


That doesn’t mean a story can’t be criticized if its own internal logic fails.

The trick is determining that internal logic. For example, a character’s abrupt shift in stance might turn on logical factors in the author’s mind, but those factors may not be visible in the story.

I’m thinking of Lee Sizemore’s final sacrifice in Westworld, which was widely criticized as pointless and out of character. (Which, frankly, may have some truth to it.)

We can’t really know why he did what he did, so we’re left to either criticize the story point for leaving us out of the loop or to accept it as what happened (for reasons we can only guess at).

It does present an interesting borderline case, though. It can be argued as shoddy storytelling, but it can also be accepted. It kinda depends on the viewer.

(I’ve raised the idea that, at that point, he may have been very hungry and light-headed. We never saw any of them eat.)


I try to be fairly accommodating in terms of character and plot.

If I can find any logical path to a plot point, any way to say, ‘well, maybe it happened this way,’ then I’m inclined to accept it.

It’s when I can’t find any way to make sense of the plot that I object.


As for that Doctor Who episode, which is widely considered one of the weaker ones this season, I didn’t think it was that bad the first time around, and a second viewing only made me like it more.

Maybe not as good as other episodes, but I’d stack it favorably against a great deal of TV past and current. (It may be that the disdain for it comes from Whovians having such a high bar.)

Stay interesting, my friends!

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

4 responses to “Story Probability

  • SelfAwarePatterns

    My issue with the episode wasn’t so much its implausibility as the fact that it seemed to leave threads dangling. What happened with all the spiders that were all over Sheffield, such as the one in the apartment? But maybe this was handled with some kind of nuance that I missed?

    The episode also felt like an excuse to make a statement about Trump. Now, I think Trump is a vile human being, but the obvious stand-in character presented was such an over the top caricature that it felt more like some kind of liberal venting fantasy. (I say this as someone who is generally liberal.)

    Overall, I think the script could have used a few more of those repetitions 🙂

    • Wyrd Smythe

      “My issue with the episode wasn’t so much its implausibility as the fact that it seemed to leave threads dangling.”

      Ha, yeah, you’re hardly the first to ask: Hey, what about all the spiders?!

      One thing that occurred to me re-watching it is that the episode seems (more than others) to lend itself to continuation, particularly with regard to the Chris Noth character. (Noth has mentioned he’d be up to return. But so has Alan Cumming, and I can’t help but see a reprise of King James I as wishful thinking. I don’t see a hook for it. OTOH, it would be hysterical for this The Doctor to run into Queen Elizabeth… her wife!)

      “The episode also felt like an excuse to make a statement about Trump.”

      I’m curious, why an “excuse” rather than, for instance, a “burning need” to express an opinion about someone who has disrupted politics and the world? Or just adding a voice to the conversation, going on record?

      (My own feeling is that we all should be screaming bloody fucking murder at the top of our lungs until this fucked up shit stops. I honestly see this as a sign the human race has no future unless we grow past this. That anyone becomes at all complacent disturbs me a lot.)

      I’m not trying to get on your case here; I’m genuinely interested in the reactions people have on this aspect. It’s the emotional freight on “excuse” and “liberal venting fantasy” that fascinates me, because that’s a take on the material. (Unless, of course, one has inside knowledge!)

      It’s a take I’ve seen a lot, and one I’ve shared myself. It’s only by reading reviews with a 180 take on the same things that spins me around and makes me — ha! — second-guess where I’m coming from.

      We crave “gritty realism” (I just watched the first season of Happy!, which I thought was hysterical, but holy canole, batman!) but object to real politics. Realism but not reality. With Who this season, a key objection (once distilled) seems to involve bringing too much real world into a wonderful fantasy.

      I think it comes from a need to escape, which is kinda why entertainment in the first place, and especially why fantasy. So bringing Trump into Who, or bringing too many specific real world issues into Who, risks ruining the escape. (Just imagine an obvious Trump character in Star Wars!)

      (My own sense we should all be jumping up and down about this might make me more accepting of that element intruding into storytelling. On some level I don’t understand why anyone is acting normal! It’s so not business as usual.)

      Sometimes I wish I could sit down with the script writers (for a lot of reasons!) and hear why they made the choices they made. Was time too tight to get it right? Budget constraints? Were choices forced on them by the network? Or did they tell the story they wanted to tell, have reasons for the choices, and we just aren’t seeing it? Love to know what’s in their heads!

      • SelfAwarePatterns

        On why an excuse, I think my issue is that I’m roundly sick of Trump, of hearing about him and his antics everywhere I look.
        Many people and sites that used to write about interesting stuff now mostly write about Trump. Our society has developed an obsession around this narcissistic clown.

        And it isn’t just Trump himself. He manages to bring out the worst in everyone, even the people I generally agree with. He often succeeds in dragging the opposition into trying to match his petty and vindictive BS, which unfortunately, is a trap I think this episode falls into.

        Unfortunately, all this focus on his personality and depravity doesn’t deal with the reasons he has a following, which just means that when we finally get rid of him, there’ll just be someone new to claim those followers.

      • Wyrd Smythe

        That very well sums up a lot of my concerns about that clown. In particular how he’s not a cause but a symptom, and we’re not talking about the causes hardly at all. (Heh, and I’m quite guilty of letting my feelings about the whole situation generate a lot of content here. OTOH, it’s like having a broken leg… it’s hard to think about anything else until you deal with the leg.)

        I suspect part of the problem is that truly dealing with the situation requires a bit too much uncomfortable self-analysis and change from all sides of this. Where we’ve taken our culture needs some examination and reform.

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