Saturday, December 3, 2016

The Monty Hall problem: Amy Santiago is right. You should switch doors.

Oh Amy, you're such a funky cat with your feisty stats, and I love you for that.

Let’s talk about the Monty Hall problem, as featured in the most recent episode of my favorite TV show of the moment, Brooklyn Nine-Nine.

In the episode, the precinct’s captain, Captain Holt, is in an argument with his husband over the Monty Hall problem, and one of the detectives, Amy Santiago (who, incidentally, is my fictitious dream woman, because she is wonderful and adorable and I am apparently a dude who crushes on TV characters), is trying to figure out a way to explain the correct answer.

The Monty Hall problem, named for the original Let’s Make a Deal host Monty Hall*, is as follows: You’re on a game show trying to win a car, and the host asks you to pick one of three doors. Behind one of the doors is the car; the other two contain a goat. After you pick a door, the host—who knows which is the car door and which are the goat doors—will open an unpicked door revealing a goat and ask if you want to stay with the door you picked or switch to the unopened door you didn’t pick. Should you switch doors?

The answer is indisputably yes, you should switch. By switching, you double your chances of winning the car—staying gives you a 1 in 3 chance of winning the car; switching improves your chances to 2 in 3. It may be counterintuitive, but you can test this out by playing repeated simulations of the problem, either in real life or online. The result, with enough simulations, will be the same: if you stay, you’ll win the car about 33 percent of the time; if you switch, you’ll win about 67 percent of the time.

Holt is unconvinced, thinking that, once it’s down to two doors, it’s simply a 50/50 chance and switching doesn’t affect anything. Amy, who’s the sort of person who gets excited over weekend math conferences called Funky Cats and Their Feisty Stats (again: fictitious dream woman like whoa), is determined to explain why switching is the correct strategy.

Unfortunately, we never get to hear Amy’s explanation because it turns out the spat wasn’t really over the math problem but rather a dearth of boning. So let me take a stab at it.

In the Monty Hall problem, there are exactly nine scenarios that can play out: you picking Door 1 and the car being behind Door 1, 2, or 3; you picking Door 2 and the car being behind Door 1, 2, or 3; and you picking Door 3, and the car being behind Door 1, 2, or 3. They're organized in this chart:

As the chart indicates, you win the car by staying in only three of the nine scenarios. You win the car by switching in six of the nine scenarios. Thus, staying yields only a 3 in 9 (i.e., 1 in 3) chance of winning; switching yields 6 in 9 (i.e., 2 in 3) chance of winning. Thus, you should switch—it doesn’t guarantee the car, but it significantly improves your chances.

Another, perhaps less intuitive, way of thinking about it: let’s say you pick Door 1. When the host asks you to switch, he’s asking if you’d like to change your pick from "Only Door 1" to "Either Door 2 or Door 3"—in other words, asking if you’d like to change your chances from "1 in 3" to "2 in 3." The chance of your door being a car doesn’t magically increase once the goat door is opened; it remains 1 in 3, so the alternative choice is 2 in 3.

It becomes a little bit clearer if, instead of three doors, we’re playing with a million doors. When you first pick a door, you have a 0.0001 percent chance of winning a car. When the host opens 999,998 doors revealing 999,998 goats, you’re left with your door and, say, Door 784,912. You intuitively know to switch—it’s highly unlikely that you picked the right door on your first try, and Door 784,912 just looks so appealing sitting there all by itself. In essence, the offer to switch is asking if you’d like to stick with your 0.0001 percent chance of winning, or if you’d like to take the 99.9999 chance that your initial pick was wrong. Even more simply, the offer to switch is asking, "Do you think the car is behind the one door you picked, or any of the other 999,999 doors?"

With a million doors, switching makes sense. And the math stays true with 500,000 doors. And 10,000 doors. And 50 doors. And indeed, all the way down to three doors.

If the whole doors thing makes this a bit opaque, consider this mathematically identical problem: let's say I'm thinking of a (whole) number between 1 and 1,000,000. Once you pick a number, I tell you that the number I'm thinking of is either the number you picked or 784,912, and I ask you if you want to switch to 784,912. Of course you'd switch—intuition dictates that it's highly unlikely you just happened to pick the correct number out of a million, and switching just makes sense. Clearly, once you're down to two numbers, switching does affect your chances significantly; you're going from having a 1 in 1,000,000 chance to a 999,999 in 1,000,000 chance. The concept remains the same if I'm thinking of a number between 1 and 1,000,000 (99.9999 percent chance of winning if you switch), 1 and 10 (90 percent chance), 1 and 5 (80 percent chance), and, yes, 1 and 3 (about 67 percent chance).

Finally, if all of that is still unconvincing, here's maybe the simplest explanation. If you stay with your door, the only way you can win the car is if you initially picked the correct door, which has a 1 in 3 chance of happening since there's only one correct door. If you switch, the only way to win is if you initially picked a wrong door, which has a 2 in 3 chance of happening since there are two wrong doors. Basically, you probably picked the wrong door initially, so staying means you're probably going to stay with the wrong door, while switching means you're probably switching to the correct door.

In conclusion, you should definitely switch doors because Amy is right. Also, Amy is totally dreamy, even when she's shame-eating hamburgers.

God, I need a girlfriend. Or Season 3 on DVD, either one's fine.

*It’s worth noting that, on Monty Hall’s Let’s Make a Deal, this situation as described never came up on the show, and thus, the name of the problem is kind of a misnomer. It's like having a problem about funny jokes, and calling it the Big Bang Theory problem. Bazinga! (For real, that show sucks.)

On the other hand, Monty Hall himself was able to come up with the right answer to the problem pretty quickly—a lot faster than many PhDs and other math experts, in fact—so maybe he deserves to have the problem named after him after all.

Thursday, September 15, 2016

The Hey Arnold! Theory of Asymmetrical Affection

Hey Arnold!, the classic Nickelodeon animated TV series about a kid growing up in the big city, gave us many gifts during its witty and often melancholy five-season run, not the least of which is what I like to call the Hey Arnold! Theory of Asymmetrical Affection.

The theory comes from “Arnold & Lila,” an episode in the show’s third season. In it, a series of Three’s Company-esque misunderstandings results in Arnold realizing he has feelings for his friend Lila, who, by the end of the episode, no longer feels the same way about him. (“I really admire you, and I treasure our friendship ever so much,” Lila says, as a way of letting Arnold down softly. “We can still be good friends, can’t we?”)

Afterward, while talking to another one of his friends about what happened, Arnold makes this astute observation:

It’s funny. When you like someone, and they don’t really like you back, it’s not so bad. But when you really like them-like them, but you find out they just like you, it hurts.

Here’s the theory in dank meme form, which I definitely didn’t make when a girl definitely didn’t turn me down for a date, because I’m definitely not a grown-ass adult who turns to late 1990s cartoons to seek comfort and wisdom when I’m feeling bummed.

It’s worth breaking down why this is true. When someone rejects you outright—not only do they not want to go out with you, but they don’t want anything to do with you—it sucks in the moment, but it’s easy to console yourself. They’re rejecting me because they don’t really know me, you can think. If they knew all the things that make me unique, if they knew my personality, my sense of humor, my quirks, then maybe it’d be different. In short, they just don’t have enough information, so they’re not rejecting me; they’re rejecting their incomplete idea of me—which, again, still kind of sucks, but you don’t take it too personally.

But when someone “partially” rejects you—that is, they like you, but they don’t like you-like you—that shit stings. You can’t hide behind the idea that they just don’t know you, because they obviously do: they like you! They see what you have to offer, and they totally appreciate it! They want you in their life—but not in that way. For whatever reason, it’s not quite enough for them to return your feelings. Why Arnold is right when he says it hurts is simple: they’re not rejecting you because they don’t know you; they’re rejecting you precisely because they do know you. (And double ouches if, in your heart of hearts, you know they’re making the right call—but that’s a different discussion for another day.)

The Hey Arnold! Theory of Asymmetrical Affection is often cited as evidence of the insidiousness of the “friend zone,” which is annoying for two reasons: the feelings Arnold is describing are a little more nuanced than the friend zone, and, more significantly, the friend zone is bullshit.

* * *

When I was in high school, the friend zone referred to the belief that if you’ve been friends with someone for too long, a romantic relationship was impossible. Nobody ever seemed to have the exact figures for how long was too long, probably because it was, again, bullshit.

It’s not that by being “just friends” for a week, or a month, or whatever other arbitrary length of time, you magically become undateable; we had several counterexamples all around us of people who were good friends first, then started dating. Instead, what’s happening is that, in that time in which you were just friends, your friend learned something about you, your personality, or your beliefs that made you unattractive to or incompatible with them—or, perhaps, something that seemed charming or intriguing at first became less so after repeated exposure to it. There’s also the harsh possibility that many find difficult to face: your friend was never attracted to you in the first place and at no point were you ever a romantically viable prospect.

Curiously, the idea was embraced by both guys and girls, with guys generally being the friend zoned and girls being the friend zoners. (The friend zone seems to be a pretty heterosexual—and, indeed, heteronormative—concept.) For guys, this provided a way to save face when he was rejected; it’s not that he was unattractive, but rather, the inviolable laws of the friend zone are what they are. For girls, this provided a gentle way to reject a guy—either because they wanted to avoid upsetting the guy by offering a real reason (“you too ugly, bro”), or occasionally because they didn’t want to think of themselves as the type of person who’d reject a guy based on, say, the fact that he too ugly, bro (“I’m not shallow; it’s just the friend zone”). It was an exercise in self-delusion, with both guys and girls having incentives to perpetuate it.

That, obviously, is less than ideal. (Are we enabling guys whose egos are so fragile that they have to comfort themselves with lies? Are we conditioning girls to avoid expressing what they really think to make guys feel better about themselves? Are we telling girls it’s better to be nice than to be honest?) In practice, though, it played out harmlessly enough: the friend zone was accepted and unchallenged, and the rejected guy could, after licking his wounds, go back to being friends with the girl. Encouragingly, guys who were purportedly friend zoned generally didn’t blame the girl (“Damn bitch friend zoned me!”) but rather themselves (“My bad, I waited too long”), which helped make guys and girls less adversaries in dating and more fellow travelers bound by the same principles.

* * *

Of course, my high school was different from other high schools. We didn’t have lockers. Nobody went to prom, even when MTV decided to record and humiliate us. Our AP calc class was as frequently about math as it was about playing Monopoly while watching the 2000 Nicolas Cage film The Family Man, apparently the only DVD our media center had. And I’m pretty sure at least a part of our campus was a repurposed Waffle House, which would explain both the smell and the despair.

This wasn't a scientific poll, but I'd say it captured our collective sentiment nicely.

So I’m not sure if I grew up with a different understanding of the friend zone, or if the definition has changed. But now, when a man complains about the friend zone, it usually takes on a much creepier vibe: I was so nice to her, and that bitch friend zoned me! She was just leading me on! Typical woman friend zoning me because I’m nice, yet she’ll probably go out with some asshole who won’t treat her right!

Obviously, some caveats: There are women who do lead men on intentionally and manipulate emotions for their own benefit (hey, free dinner!), because women are people and some people are shitty. And yes, there are women who find niceness unattractive and assholery attractive, because women are people and some people have weird issues. But I feel confident that these cases don’t represent the majority of friend zone whining.

(Also, as an aside, my pet hypothesis is that being nice isn’t so much an inherently attractive thing, but rather an intensifier if you already find someone attractive. Like, niceness won’t do much if someone doesn’t think you’re cute, but if you’re cute and it turns out you’re nice, it makes you that much cuter. It is in this way that niceness is a lot like eyeglasses, tattoos, and impeccable grammar.)

This mindset comes from the idea that if a man does enough nice things for a woman, he’s entitled to her time, her affection, or her body. In this view, a woman isn’t a person with her own desires, but rather a puzzle to be solved and a prize to be won. Or not even that—she’s a product to be purchased, which he deserves because he paid for it with his niceness. This is a point rebutted by an oft-memed quote misattributed to Sylvia Plath:

This betrays a serious lack of empathy. All of the men bemoaning how nice guys get friend zoned would disagree that, if a woman he found repulsive baked him enough cookies or gave him enough gifts, he should ignore his own desires because she earned him and that overrides his feelings. All of these men would recognize that they get to decide with whom they go out and have sex, and “because she’s really, really nice and really, really likes you!” isn’t sufficient for them to set aside their own preferences.

But they generally don’t extend that same courtesy to the women they want to bone. And I get it—you’re the main character in your own story, and she’s your perfect dream girl. Super Mario Bros. taught you that if you do the right sequence of things, you earn the right to some Tanookie Mario with Princess Toadstool, even if she has expressed no interest in you and your shoes smell like squished Goomba.

However, she’s the main character in her own story, too, and just like you want someone you think is gorgeous and lovely, so does she. And you might not be that person, no matter how nice you are. It’s okay. It happens sometimes. You just have to deal.

* * *

One of the worst things about the friend zone is that it makes some men embarrassed of their friendships with women. Being friend zoned is seen as a mark of emasculation, and since there’s supposedly no other purpose of being friends with a woman besides having sex with her, every platonic friendship with a woman is viewed as an implied rejection and thus a failure. Every act of kindness shown to a woman that doesn’t result in her ripping off her clothes isn’t just a waste but a pathetic act of self-torture (“You picked her up from the airport? FRIEND ZONE LEVEL 100! She had a bad day at work and you listened to her for a few minutes without getting any ass? FRIEND ZONE LEVEL INFINITY!”). These men will have fewer friendships with women, which means they’ll have fewer women they care about, which means they’ll be more accepting of whatever misogynistic nonsense they read on some creepy subreddit. And the cycle of toxicity will self-perpetuate.

The reverse happens, too. If women are constantly suspicious of their male friends’ motives, they may decide that male platonic friendships are more trouble than they’re worth. Who wants to deal with the hurt when someone you thought was a really great friend just wanted to have sex with you? Who wants to be accused of leading someone on or berated for being a typical woman who can’t appreciate a nice guy?

(Also, what’s the deal with people saying that if a man is sexually attracted to a female friend, it’s proof that he’s not really her friend, or that male-female friendships are impossible? If a woman is attractive, there’s a decent chance that at least some of her male heterosexual friends would be pretty stoked to sleep with her. But if she doesn’t want to sleep with them, it’s entirely possible that they’re okay with that and value and appreciate her friendship anyway. In fact, it’s kind of fucked up that “is sexually attracted to a woman” and “genuinely values a woman’s friendship” are often accepted as mutually exclusive concepts—what the hell does that say about us? In any case, as the great Mayor Diamond Joe Quimby says, it can be two things.)

* * *

During the rest of Hey Arnold!’s run, Arnold still crushed on Lila. And Lila continued to view Arnold as purely a platonic friend, much to his chagrin. But Arnold didn’t accuse Lila of leading him on or complain that Lila was a stupid bitch who friend zoned him because she couldn’t see how nice of a guy he was.

Yes, asymmetrical affection sucks for the reason Arnold articulated perfectly, but Arnold found a way to deal with it without being a self-pitying asshole. As with most things in life, we should all be like Arnold.

Or maybe Gerald. That dude has a field named after him and has some pretty rad hair. Let’s be like him, too.