Four of Wands, Page of Math

Like many people, I have an embarrassing hobby.

So I'm a statistician, right? A general fan of science and facts and stuff. A stickler for evidence. The kind of person whose mantra is correlation does not necessarily imply causation, except when the causal relationship in question isn't based on quantitative data and therefore can't really be described using the term "correlation," in which case the dubious argument is better criticized by referencing the logical fallacy "post hoc ergo propter hoc." (Because I'm also the kind of person who is a pain-in-the-ass stickler for using the word "correlation" correctly.)

So, it follows that I wouldn't be into any sort of mystical divination practices, since their fakeness is fairly obvious. Even my homegirl Hermione rolls her eyes at them. She trusts the judgment of a telepathic singing hat, but not someone's subjective interpretation of a pile of tea leaves, because even literal witches know that divination is bullshit.

That is my embarrassing hobby: I read tarot cards. I don't believe they have any predictive power, but I know how to do the spreads and interpret them as though they do. The only excuses I have for why I read tarot cards are

1. You can convince drunk people that you're magic
2. They're pretty
3. It's kind of like Rorschach blots, you know? The meaning that you impose on the ambiguity can highlight thoughts, feelings, and motivations that might otherwise be difficult to identify
4. They're really pretty
5. Did I mention how pretty they are

they are really very pretty

My friend Lauren wants to go back to school to finish up her degree, and a while back, she asked if I could read her tarot cards with regard to her prospects. We shuffled and re-shuffled (a LOT) and I drew three cards for her: a reversed ace of wands (stagnation or lack of passion for a new opportunity), a nine of swords (mounting anxiety and insomnia), and a reversed five of swords (unavoidable catastrophic failure). Not even the most liberal interpretation could spin anything positive out of that hand.

So I blew it off and shuffled again, because Lauren is awesome and deserves a fortune-telling session that predicts piles of cash and hookers (lookin' at you, ten of pentacles). And the new cards were a reversed high priestess (impatience, wasted potential), a five of pentacles (poverty and bad luck), and the goddamned devil (which is pretty much as bad as it sounds).

This ridiculous bullshit continued for about seven more drawings, separated by increasingly intense shuffling sessions, including the foolproof throw-all-the-cards-in-the-air-and-swear-loudly method. Time and again, Lauren got cards that weren't just irrelevant or neutral, but seemed to be actively shitting on her dreams. More than once, she shouted at me, "you're a statistician-- what are the odds of this?"

Challenge accepted!

At first I thought it would be pretty easy to answer Lauren's question. How many ways can there be to draw three cards from a deck? While a card's position in its spread usually affects its interpretation, we weren't applying any past-present-future or problem-advice-outcome meanings to the three card spreads we were doing, so order doesn't matter for our purposes. There's 78 cards in a tarot deck, so there's 76,076 possible hands of three.

Now we just have to determine how many of those spreads would be unfavorable for Lauren's education! This part turned out to be trickier.

Since orientation affects interpretation, each of the 78 cards has 2 possible outcomes, for 156 total. I categorized each of these 156 outcomes as positive, negative, or irrelevant with regard to Lauren's education, because ambiguity is for chumps. Of those, there are 43 good possibilities (28%), 64 bad possibilities (41%), and 49 possibilities (31%) that have no bearing on Lauren's question (get outta here two of cups, we weren't asking about crushes). I defined an unfavorable spread as any combination of three cards that includes no positive results and at least one negative result: either all negatives, two negatives and an irrelevant, or one negative and two irrelevant.

And that's where I ran into a bit of a problem. See, I planned to calculate all the different ways one could draw any of those three hands using plain old n-choose-k, where I'd be looking for how many ways to choose 3 negative cards from the pool of 64... but there aren't exactly 64 negative cards. It's impossible for me to first draw an upright eight of swords (feeling trapped by circumstance) and then draw a reversed eight of swords (feeling trapped by circumstance, but like, even worse). But I've included both the upright and reversed versions of the card in my tally of negative outcomes. Essentially, I've backed myself into a mathematical corner where I've artificially doubled the size of the deck: I'm doing calculations based on 156 outcomes, rather than 78 outcomes with two variations each.

To be truly rigorous, I should tally up how many cards switch from positive to negative, from irrelevant to positive, from negative to irrelevant, etc, versus how many cards maintain their general meaning when reversed, and work those numbers into much longer calculations of conditional probability.

Yeah, I based my calculations on the imaginary 156-card deck, where each orientation of each card counts as its own separate card. It's fortune-telling, for crying out loud, I'm not going to worry too much about rigor.

If we allow the fudge-factor of a theoretical 156-card deck to deal with the problem of card orientation, there are 620,620 possible combinations of three cards that one can draw from a tarot deck. There are 41,664 ways to get three negative cards, 98,784 ways to get two negative cards and one irrelevant card, and 75,264 ways to get two irrelevant cards and one negative card.

All together, that's 215,712 crappy hands, out of 620,620 possible hands. There's about a 35% chance that any individual tarot reading I do for Lauren's educational prospects will be negative. If we multiply things out to reflect the fact that Lauren got eight successive crappy readings, we get a probability of just over 0.02%. So... huh. Dang Lauren, sure looks like the universe has it in for you.

Technically, the universe has it in for all of us. Enjoy your inevitable disintegration, everything! <3 Entropy

Public Radio Pledge Drive Probability

North Carolina Public Radio Pledge Drive Season is the worst time to miss a call from an unfamiliar number. I have missed several such calls in the past few weeks, and I am convinced that each one was Eric Hodge. He was calling to inform me, via vague, leading questions about famous landmarks and culturally-specific foods, that I'd won a drawing for one of the fantastic getaways they always advertise. One of these days, I'll be right. But which one of these days?

It's a little tough to figure out how likely I am to win any particular WUNC Trip Drawing. My probability of winning is always nonzero, since I'm a Sustainer, which means I'm a) better than everyone else and b) automatically entered into every Pledge Drive drawing. The amount of funding that North Carolina Public Radio receives from listener contributions is public knowledge, but the precise number of contributors isn't listed anywhere that I could find. Even if I could figure out how many Sustainers there are, the number of one-time-gifters in the pool changes with every drawing.

There are a couple ways I can estimate how many people listen to WUNC, which is a good start for figuring out how many people are competing with me for that weekend getaway in France. This Nielson report from December of last year indicates that 90% of Americans listen to radio each week, and the Raleigh-Durham region has a population of about two million, so the local radio-listening audience is probably somewhere around 1,800,000. WUNC's weekly cume is a little under 17% of the market, so there are probably at least 300,000 people tuning in each week. Maybe. Man, I hope I'm estimating radio listenership the right way, since it's sort of my job that I get paid for!

So, let's be optimistic and assume that maybe one out of every twenty listeners donates to the station. I've got no clue if that estimate is wildly high or wildly low or wildly spot-on, never having worked for an organization that depends on donations, but it feels like a vaguely educated guess. That's 15,000 donors total. It probably fluctuates, but let's say, for argument's sake, that I have a 1 in 15,000 chance in any given drawing of winning a trip to Rome. There are three WUNC Pledge Drives each year, and each one seems to have at least five trip drawings, so I have fifteen chances each year to win.

How long would I have to wait before I could be at least 90% sure of winning at least one trip drawing? Put another way, how many times do I have to enter a drawing so that my chance of losing every single last of them dwindles to 10% or less? My chance of losing any individual drawing is 14999/15000, and my chance of losing all the drawings I enter is 14999/15000 raised to the power of the number of drawings in which I participate. So let's solve this equation!

The sooner I get LaTeX on my new computer, the better

Using the definition of a logarithm and the change of base formula, both of which I totally remembered and did not need to look up just now, because I am a smart mathematician who never ever forgets really basic important facts, and being chronically rusty on logarithms definitely isn't a significant source of anxiety for me, we conclude that the number of drawings I'd need to enter in order to have a 90% chance of winning at least one of them is... 34,538 drawings, or over 2,300 years worth of pledge drives. Lucky for me, my loyalty to quality radio programming is as undying as the sun. Which means, strictly speaking, not actually undying. But undying enough to last for 2,300 years.

This one isn't about statistics, but is instead about my mom, and her death

I feel very sad, very often, and very little remedies the sadness.

I've started writing this thing so many times, and discarded so many drafts. If I took all the words I've written and deleted since December 7th 2013 and put them together, I'd win NaNoWriMo. Sometimes what I wrote was eloquent, but most of the time it was impenetrable garbage that rambled on way past the point where it stopped making sense. Sometimes what I wrote was angry, and mostly it was angry at people who had nothing to do with what I was angry about. Sometimes it had a lot of science in it. Sometimes it used a lot of metaphors. Sometimes it included a lot of fandom references. Sometimes it had pictures, and sometimes they were pictures that I drew (poorly). Sometimes I was bitter. Sometimes I wrote while I was sober, sometimes I wrote while I was drunk, and sometimes I wrote while I was crying and couldn't stop.

Mostly, all the things I wrote were just different ways of saying the same thing: My mom died. I feel very sad, very often, and there are very few things that remedy the sadness. Writing isn't one of them.

Correlation and Causation (It's Kind of a Big Deal)

My sister sent me a link to an article titled "Earn more money, when you have more sex, study says (seriously!)" Being a math teacher herself, she had some doubts about the article's claims. She linked me to the study on which the article was based and passed along her thoughts on its methods:

Is it me, or are the guy's statistical interpretations (from the journal article) not all correct? Sketchy claims include: 

"For both sexes, in Panel I, we observe that a one standard deviation increase in sexual activity increases hourly wages by 3.2%, other things being equal." 

"The importance of the sexual activity variable can also be assessed by the fact that if we regress a single wage equation without the sexual activity variable, the R^2 is 0.821, while if we consider the sexual activity variable (as in Table 5), the R^2 is 0.842. In other words, the wage estimation becomes more precise if we consider the sexual activity variable." 

Um, R^2 always goes up when you add an additional variable, that's why we use adjusted R^2?

Not to mention the implied causation claim. I am not surprised at the correlation since there's a documented positive correlation between being married and having higher income, but really "Earn more money when you have more sex"?

My sister hit the nail on the head with the R-squared thing. To quote the stats textbook I keep in my cubicle for the purpose of making my coworkers think I'm smart, "in practice, the best model found by the R-squared criterion will rarely be the model with the largest R-squared." Adding additional explanatory variables to a regression equation will always increase R-squared, whether or not those variables are truly significant. Adjusted R-squared takes into account the number of explanatory terms already present in the model, and is often a better way to determine which variables are worth retaining in a regression equation-- it helps temper the impulse to cram in as many explanatory variables as possible. I would not interpret an increased R-squared as conclusive evidence that the sexual activity variable is worth retaining in the final regression equation.

The causation claim raised my suspicions as well. It's very difficult, often impossible, to prove causation in an observational study like this one-- there are a million confounding variables out there. "Correlation doesn't prove causation" has become the go-to comeback favored by science fans the world over. It's an easy shot to take at almost any of these attention-grabbing, "studies show" headlines. It's so easy it could feel like a cheat code for sounding smart, if it weren't so often true and applicable.

This excellent needlepoint design is sadly sold out on Etsy.

Homestuck Revisited

I made a mistake.

A huge, HUGE rookie mistake.

Remember this drawing from the Homestuck Post?

It was more correct than I was about the probability of a female-female conversation occurring in a group of size N containing F females.

Was Hot Fuzz a Documentary?

An unconscionably long time ago, Abhi posted a link on my Facebook wall to a study about the comparative dangers of living in the city versus living out in the country, with the instructions "BLOOOOG! GOOOOOO!" I meant to write about it a lot sooner, but, you know. Annual family reunion/beach vacation, so like, nothing productive happening THAT week. Now that my vacation's over and my sunburn's faded enough that I can wear shirts again, it's high time for some bloggin'!

The National Geographic article is split into five "surprising" facts about country living versus city living, "surprising" mostly because they run counter to some conventional wisdom about safety. Following their lead, I'm going to use this study's findings to address three key quotes from the cinematic masterpiece on the relative dangers of small town life, Hot Fuzz.

SANDFORD'S FINEST

The Lottery Reloaded

So my company offers clients mathematical analysis of trends and probabilities. We also have an office lottery pool. Make of that what you will.