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Pizza!

I’m ridiculous, but I wanted to share this.

Last Friday, I got pizza from Domino’s for the first time since college. I’d heard that they did some soul-searching about a decade ago and discovered that perhaps customers didn’t like pizza that tasted like ketchup on a cracker topped by a slice of pasteurized processed cheese-food product, and re-engineered their offerings into something that could broadly be described as “pizza”, but I hadn’t gotten around to actually trying it. Dylan had been suggesting that he might like to since he’d seen a teacher eating some at lunch, and some coworkers were talking about their carryout special, so I pulled the trigger and bought a couple of pies, and you know what? It was pretty good. I liked it. Dylan liked it. Evie ate the cheese and pineapples off of two slices and begged for more. Leah thought it was okay.

So in celebration of this surprisingly edible pizza, I wanted to offer up the following meme:

Upon reflection, associating your pizza company’s mascot with avoidance may have been a misstep.

Misspent Youth: The Toothpaste Millionaire

There’s a handful of books I read — or more accurately, were read to me — in my youth which left enough of an impression that I’ve felt compelled to track down a copy thirty-or-so years later, particularly now that I’ve got little’uns to read them to. This is still a work in progress. Evie’s not old enough to appreciate The Frisky Kittens or Piglets at Sea yet, and I think I should give Dylan another year or two before I spring The Westing Game or From The Mixed-Up Files of Mrs. Basil E. Frankweiler on him. But we made some headway recently when, on the second try, he got interested in Jean Merrill’s The Toothpaste Millionaire.

Jean Merrill is best known for an earlier book, The Pushcart War, which I bet is pretty surreal to read now since the publishers keep pushing forward its then-near-future dates to keep them near-future, but probably they haven’t updated Merrill’s writing style to be less full of charmingly dated minutia. Also, odd coincidence, Merrill and I have the same birthday.

I don’t remember exactly when we read The Toothpaste Millionaire in school, but it was definitely during Media Center (even back then, in the misty murk of history, “Library” was deprecated). Over on the far side where there was a clearing that served as a small presentation stage, flanked by the bookcases of Hardy Boys books, sitting in a position whose old name I completely agree was inappropriate and had to go, but for which they have not yet invented a replacement name I can possibly take seriously (“Criss-Cross-Applesauce”? Come the fuck on. That is stupid. That’s worse than “letter-carrier”. Get the folks who came up with “firefighter” on this. They know their stuff).

The basic story is this: Kate McKinstrey and her family move from Connecticut to Cleveland, and the first friend she makes is a clever and economical boy named Rufus Mayflower, who helps her make saddle bags for her bike. Erelong, Rufus develops an obsession with the cost of toothpaste, and decides to go into business selling his own homemade baking soda-based dentifrice under the name brand-name TOOTHPASTE, and sells it for three cents a bottle. To say that hijinks ensue is being a bit hyperbolic. Spoilers: Rufus makes a million dollars.

The book is pretty funny. Dylan found it hilarious, in fact. It’s not, if you’re an adult, that much happens which is especially silly in itself; in fact, the whole book is a series of people acting reasonably and making rational decisions. But there’s a lot of comedy in the outsider point-of-view and the… Let’s say child-like plainspokenness. Have you ever tried to explain a complicated social concept to a child, like why we segregate bathrooms by gender or why it’s not socially acceptable for men to wear skirts or why we have capitalism, and halfway through realized that it actually is, in fact, ridiculous? That happens a lot in this book. A lot of “Well why not?”

What I noticed on this read through is that the structure of the book is a little strange. The book is largely presented as Kate recapping how Rufus solved a series of logistical problems. The narrative is sort of fragmentary because of this: each chapter is essentially a separate episode, and the linking between individual scenes is generally pretty weak. Some chapters are barely narrative at all. For instance, one chapter covers Rufus doing a TV interview. In a more traditional narrative, you might have a scene of Kate and Rufus talking beforehand, and a scene of Kate watching from the audience and her thoughts and feelings as Rufus goes on the air. But instead, we get a chapter that’s mostly exposition, with Kate dryly explaining about the local talk show, a transcript of the interview itself, and a coda talking about all the new orders they got, including an anecdote about an order from someone in California who’d been on the phone with a relative in Cleveland when the interview aired.

The style works for me, and I think it would work for most kids with engineering-type minds, who’d be more interested in the problem-solving than in any sort of character-driven narrative. Even in the ’80s when I first read the book, there was a weird uncanniness to the ’70sness of it too. Kate accidentally buys five gross of toothpaste tubes thinking she’s buying five dozen, though it still only ends up costing her like five bucks. Which is a lot of money back then.

But having said that, I find that my memories of the book aren’t weighted proportionally. Virtually all the conflict in the book comes in the last fifteen pages, and it feels almost entirely offhand, even though my memory tells me it’s a major theme (My memory also tells me that Kate also bought a barrel of aglets on a lark. This didn’t happen at all, and now I’m wondering if maybe that’s something I am remembering from a different book? Anyone recall a book where the narrator and the protagonist go to an auction and the narrator buys a barrel of aglets?). Another thing I remembered from my youth was a pervasive sense of Rufus being a little “off”. Back in elementary school, I wouldn’t have had the concept, but Rufus, with his obsessiveness over waste, his fetish for honesty, and his frequent displays of frustration at people acting like people, definitely struck me at the time as being on the spectrum. Reading it now, I didn’t get that impression at all, and I’m curious where it came from, whether I was projecting from something else, or maybe it was something I was nudged into by the teacher? The whole of the conflict is squeezed down to just a couple of chapters, the main one of which is deliberately gonzo.

After the major toothpaste brands start folding as a result of the ensuing price war, the remaining players invite Rufus to a “conference” about the health of the industry. It turns out to be an attempt to involve him in a price-fixing scam, which ends when it turns out that the FBI just happens to be in the next room over and promptly arrests the heads of the competing brands. Kate presents this in the form of a screenplay, having decided that she’d like to be a screenwriter specializing in torn-from-the-headlines dramas. And then the chapter ends with the offhand reveal that the toothpaste factory was blown up by a gangster working for the mob.

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Kinda moving…

So apparently Google’s so hell-bent for everyone moving to https that they’re going to start popping up warnings in the near future if you go to a non-SSL page.

This prompted me to buy an SSL certificate. But unfortunately, my web host sucks ass and was able to confuse me into buying a certificate which only covers www.trenchcoatsoft.com, and not blog.trenchcoatsoft.com.

Well, actually, I think they manipulated me into buying both, but they only gave me the first one. They’ve agreed to refund the money, but they’re now claiming that they don’t even offer ssl for subdomains.

So I decided that, since I’m not really using my main domain for anything (I’ve got some ancient software up there, which I’ll make available again if anyone complains), I’ll just shove the blog over to the main domain. Which means that from now on, you can get to this blog, with all the safety and security that a green lock up in the address bar implies, at its new URL, https://www.trenchcoatsoft.com. The old url, http://blog.trenchcoatsoft.com should still work as well, and any deep links you’ve got on hand will work as well, just unencrypted.

So you may now visit in the happy knowledge that evil haxors will not be able to intercept you as you read my ridiculous blog about failed ’80s TV series.

I don’t believe it’s so.

The next couple War of the Worlds posts are already scheduled so they’ll probably run on time unless I decide not to, but I don’t know if I’m going to keep going after that. I no longer enjoy writing about apocalypses now that I live in one.

But I wanted to share a few lines from a song that would have been featured in today’s post if the human race hadn’t just proven that they deserve the extinction that’s coming faster than ever now.

 

Once, there was a time when I believed,
Without hesitation,
In the power of love and truth to conquer all,
in the name of salvation.
Tell me what kind of weapon is love, when it comes to the fight?
And how much protection is truth, against all Satan’s might?

 

Hate wins.

I am done with you, my fellow Americans. Fuck you all.

Y’know what? I’m white. I’m straight. I’ve got money. I am going to be hurt less than a lot of people. So as Emperor Trump destroys the economy and you motherfuckers have to explain to your starving children that you voted for a rapist because of something to do with emails that you don’t even understand yourself, when you’re kowtowing to your Russian overlords as the KKK drinks to their triump, I’m going to just enjoy the fucking schadenfreude of knowing that you motherfuckers got the president you wanted. I’m gonna laugh my ass off when you fuckers in Florida are drowning because that “Chinese hoax” has submerged your state. We offered you equality, and you wouldn’t take it. There’s nothing left I can do to help my gay friends, my muslim friends, my transgender friends, my hispanic friends, my female friends from the pain that’s been unleashed on them, They will get the worst of it, but I promise that you will burn too. All that’s left for us now is to make sure that when you burn us down, you burn with us. Fuck you all.

I’m sorry Dylan. I’m sorry Evelyn. We failed you. I failed you. The human race failed you. It’s not right and it’s not fair and I don’t know if you’ll be able to fix it. If the world even still exists when you grow up, you should curse us and spit on our graves for what we did to you. I am so sorry. Don’t forgive us. Don’t ever forgive us.

 

Monty Hall and the Door of Probability

I felt like writing something else this week. As it happens, no episodes of War of the Worlds aired during March of 1989 or 1990, and I’d planned to insert a gap of a few weeks in my own schedule there, to cover things like the Asylum movie and ALF and maybe the final episode of Webster. But there’s two more episodes of season one and one more of season two before that happens, and I feel like digressing now instead.

The Monty Hall Problem is a probability puzzle that’s in a class of problems that are unusual in just how ridiculously hard it is to make your brain believe the right answer despite how simple it is. When columnist Marilyn vos Savant presented the correct solution (With the caveat that the specific phrasing of the puzzle she used contained a significant point of ambiguity) to the puzzle in a 1990 column for Parade magazine, she was inundated with thousands of letters — many from PhDs — insisting she was wrong. Many of the writers were real assholes about it. Ridiculously famous mathematician Paul Erdős, famously, refused to believe the correct answer, even with formal mathematical proof, until he saw the result verified in computer simulation. I happened to overhear some people discussing the puzzle recently, and as usual, someone got it wrong and could not be convinced of the right answer.

The puzzle is this:

You are a contestant on $FAMOUS_GAME_SHOW. The host, $WELL_KNOWN_TELEVISION_PERSONALITY presents you with a choice of three doors. Behind one of these doors is a BRAND NEW CAR(!), while behind the other two are goats. You are asked to select one of the three doors. After you have made your selection, the host opens one of the unselected doors, revealing a goat. He offers you a choice: open the door you initially selected and claim the prize behind it, or switch your selection to the remaining closed door. What course of action should you choose?

Picture courtesy of Wikipedia If you haven’t already heard this one, you almost certainly think that it makes no difference whether or not you switch: the car is equally likely to be behind either door. But given that, whether or not you possess the comparatively rare gift of statistical literacy, you almost certainly possess basic English literacy to have worked out that if that were the answer, this would hardly be a famous brain teaser.

The answer, of course, is that you should always switch. The probability that switching is the right strategy depends on certain assumptions not present in the way I phrased the question, but there’s actually only one non-standard assumption you can make where switching is a bad idea (The one where Monty is an asshole and wouldn’t have opened the door had your initial choice been incorrect). Well, okay, switching is also a bad idea in the non-standard assumption that you’d rather win a goat than a car. Fair enough. My family already owns two cars, but zero goats. Some people think that clarifying the standard assumptions also makes the analysis more intuitive, but I don’t: the assumptions primarily impact the question “What’s the best general strategy on this game show?” rather than the question-as-asked, “What do you, the player, do next?”

These assumptions, for clarity, are:

  • Monty always opens a door and offers a switch
  • Monty never reveals the car when he opens a door
  • If the player’s initial choice is the car, Monty chooses the door he will open at random

That third one in particular I think just makes things more complicated. It’s necessary to close a loophole where you might be able to deduce some extra information from which door Monty opens: if Monty has a preference for lower-numbered doors, then Monty opening door number three means you should absolutely switch — but, crucially, you should still switch even if he doesn’t. The other two conditions don’t actually apply in the specific case specified, since Monty did open a door and didn’t reveal a car. Removing them can reduce the probability that you should switch, but not below 50:50.

That’s why those assumptions are so often claimed to make the problem easier to understand. But the thousand letters to Marilyn don’t bear this out: most of the letter-writers intuited those constraints but still got it wrong. Marilyn herself suggested that the problem would be more intuitive if you cranked the numbers up: make it a million doors, and have Monty open 999,998 of them. Do that, and the magical power of big numbers makes your brain change the question it’s asking from “Which door is the car behind?” to “Did I choose right the first time?”. But it turns out that when you increase the numbers, people are more likely to switch, but they still don’t believe the probability changes.

I’ll spare you the formal proof of the answer for the moment. The reason I’m writing this is that I wanted to share with you how I came to believe the answer. The math of the answer, I got right away. The problem was, it didn’t feel right. It felt like a glitch in the Matrix, or like Scotty’s solution to the Kobyashi Maru: a weird math thing that didn’t map to reality, like imaginary numbers. Or real numbers, for that matter.

I had a hard time explaining this to people. And by “people” here, I’m taking some liberties, because I mostly mean “mathematicians”. They couldn’t read it any way other than, “I don’t understand the math.” But I understood the math. What I couldn’t understand was reality.

But that million-door variant of the problem reminds me of another probability question I used to have a hard time with. The flipped coin question. Imagine you have a fair coin, and you flip it nine-hundred-ninety-nine thousand, nine-hundred-ninety-nine times, and it comes up heads each time. What are the odds that it’ll come up heads on the next flip?

I find it completely intuitive that the answer is ½. And when you phrase it the way I did, I hope most people agree. But there’s a whole family of similar questions where the math is exactly the same, but people will fall prey to a gambler’s fallacy and become convinced that the coin is due. It’s all about the law of averages: a tossed coin coming up heads a million times in a row is astronomically unlikely. One more heads flip, and it will happen. Therefore, the odds of getting that last heads flip must also be astronomical.

That’s ridiculous, of course. But the usual explanation for why it’s ridiculous never felt quite satisfying to me: “Because how could it not be 50:50? It’s not like those past coin flips have some magical power to change the probability of the next one.” It’s true, but it doesn’t address the “law of averages” thing that makes it feel wrong.

The answer that satisfied me is this: your belief that getting a million heads in a row is astronomically unlikely is arbitrarily privileging the number “a million”. Because it’s big and round. Because you know what else is astronomically unlikely? Flipping heads nine-hundred-ninety-nine thousand, nine-hundred-ninety-nine times in a row. And you’ve already done that. The punchline is this: how much more astronomical are the odds of a million heads compared to the odds of nine-hundred-ninety-nine thousand, nine-hundred-ninety-nine? Exactly twice. The way I look at it is this way: you’ve already used up most of the “unlikely” getting the first nine-hundred-ninety-nine thousand, nine-hundred-ninety-nine heads. And the “amount” of “unlikely” you’ve got left to get to a million is exactly fifty-fifty.

Credible mathematicians would say I’m being obtuse. It’s fifty-fifty because each coin toss is an independent event with independent probability and that’s the end of it. But there’s the rub, right? Why is each coin toss “independent”? Mathematically, each coin toss is independent because the probability of flipping heads on flip N+1 is the same regardless of the outcome of flip N. But this is just begging the question (And for possibly the only time in my life, I think I am using the phrase “begging the question” in its pedantically correct sense): the odds are the same because the flips are independent, and the flips are independent because the odds are the same. Remove the abstraction of mathematics, and you get at a more reasonable explanation: the flips are independent because a coin, being a slug of metal, neither remembers nor cares what the outcome of the last flip was.

What always made me uncomfortable, then, was the context switch: the simple answer can be expressed as elegant, pure, abstract mathematics, right up until you get to the bit where we simply have to take for granted that coin tosses are independent events. It just feels wrong for math to throw up its hands at that point and say, “Because it’s a coin, dumbass!” Especially when you frame it in a context like, “We’re one flip away from a million heads in a row,” because “A million heads in a row” sure doesn’t sound like an independent probability. On account of it isn’t: P(one million) = 2-1,000,000; P(one million|999,999)=2-1.

And indeed, once I made the mental switch to the notion of having “already used up” the improbability, the whole thing made a kind of actual sense to me, rather than just being a series of mathematical formulae that I knew on account of I’d read them in a book and remembered them. That coin isn’t due: it whittled away at the improbability slowly, over the course of the previous almost-a-million flips. Put another way, if you do the coin-flip experiment in 21,000,000 parallel universes, there’s only one where you get heads a million times, and many where you don’t — but if you know that you got the first 999,999, then you aren’t in most of those universes: you’re in one of two (Or maybe both of two. Quantum mechanics is weird).You can apply that paradigm to other things as well. Larry Niven once famously declared that an ear of corn was more likely to father a child with Lois Lane than Superman was, meaning that out of all possible universes, a Kryptonian would be sufficiently genetically similar to a human for cross-breeding to be possible in an infinitesimal percentage of them. But this fails to consider that it’s already an infinitesimal percentage of said universes where Kryptonians would be sufficiently similar to humans that Lois would even be willing to try (You may, as the mood strikes you, include or exclude those universes where Lois is just Into That Sort of Thing). It’s not “tiny number out of nigh-infinite number”, it’s “tiny number out of very-slightly-less-tiny number”.

Let me get back to Monty Hall. I slipped the key observation to the problem in a few paragraphs back without calling attention to it. Did you catch it? People get the Monty Hall problem wrong because they don’t approach the question that was asked. Most people hear the problem and approach it as “Which door is the car behind?” But that’s not the question that was asked: the question was “Should you switch?”, which is to say, “Did you pick the right door on the first try?” The answer to that question is very obviously, “Probably not.”

Where I always got hung up was the idea that the solution lay in what you knew when you made your initial choice, versus what you know afterward. People kept trying to explain it to me that way, and it just sent me astray, thinking that the glitch in the Matrix lay somewhere in the vicinity of, “Had I known it wasn’t door number 3 when I chose door number 1, I’d have chosen door number 2 instead.” But that’s nonsense.

Here’s the magic thing, and why I don’t think that restating the standard assumptions about Monty’s behavior actually helps people understand the paradox. What Monty does has no impact on the input to the question. Monty isn’t giving you actual new information you can use to make a better guess, as people think. If he were, then it reduces to just making a new guess and this time it’s out of two rather than out of three, and, yeah, the odds are even. But what Monty is actually doing by opening a door is nothing more than distracting you from the actual offer he’s made: switch, or not-switch.

Here is when I finally “got it”, when I finally moved beyond understanding the math to actually seeing how the physical reality of the problem worked, and stopped thinking it was some kind of weird Schrödinger’s goat thing. Let’s just remove Monty from the equation altogether:

You are a contestant on $FAMOUS_GAME_SHOW. The host, $WELL_KNOWN_TELEVISION_PERSONALITY presents you with a choice of three doors. Behind one of these doors is a BRAND NEW CAR(!), while behind the other two are goats. You are asked to select one of the three doors. After you have made your selection, the host opens one of the unselected doors, revealing a goat. He offers you a choice: open the door you initially selected and claim the prize behind it, or switch your selection to the remaining closed door bet against yourself: your door will remain closed, and instead, you’ll get what’s behind both other doors. What course of action should you choose?

I submit that this variation is mathematically identical to the standard problem (plus or minus one goat). All Monty’s doing when he opens one door is finding a misleading way to explain that your choices are between “Get the car if you picked right initially” and “Get the car if you picked wrong initially”. That’s all you’re being asked. And there’s no variation of the assumptions where what you’re being asked it to “pick again with new information”: you’re only ever being asked whether you think your first guess was right or wrong. In some of the variations, Monty’s behavior might be encoding a hint about the answer to that question, but it’s never changing the question. For all mathematicians like to say that the problem is people not understanding probability, it’s actually a problem of people not understanding the question.

Remove the obfuscating detail of Monty opening one door and offering to let you have the other, and suddenly, it all becomes clear: the second choice is not to pick a door, but to reject one. That’s why switching wins two-thirds of the time.

And hey, worst case, you get two goats.

Image by Mark Katzenberger. Some rights reserved

Image by Mark Katzenberger. Some rights reserved

Program Note

By popular request of her mother’s doctors, my daughter has rescheduled her arrival for tomorrow morning. There may be some interruptions to the normal programming schedule over the next few weeks.

New Doctor Who posts will still be dropping on Saturday at least through the beginning of April, but there might be some slippage after that. Tomorrow’s War of the Worlds post has been bumped back to next week for a special post.