I'm thinking of a number between zero and zero

In response to a recent post in which I claimed that the male to female ratio for Americans between the ages of 25 and 33 is 9:1, reader Maris whined pondered aloud:

so what the hell is my problem then? the ratio is 9:1 and I still can't find a decent man to date? Perhaps my standards are just too high- but you tell me. There is a job requirement (you must have one), you must not be married (seriously), and live in the general vicinity of where I live (this is only a recent requirement), have a strong dislike of the packers and be a cubs fan (this is non-negotiable) and still nothing! Do I stay here and hedge my bets (the guy sitting across from me in the coffee shop (early 30's) is rockin a tee that poses the question "wanna ride me longboard?" Is this really what I have to look forward to in this town? I like the longboard as much as the next lady, but do you really need to ask me about it publicly and in group format? Please. Or head for the big city where the ratio of douchebags to non douchesbags is just a bigger number and I get to hear/see more inappropriateness. Such a pressing decision. Perhaps I'll just get another dog.

We have a lot to tackle here.  But, before trying to answer Maris's questions, I have one for her:  is the quote "wanna ride me longboard?" supposed to be "wanna ride MY longboard?" or does the tee feature a leprechaun posing the question, in which case the spelling is correct?

Let's do a naive statistical analysis to see how many males meet Maris's requirements.  First, she informs us that her man must be employed.  Let's say that the unemployment rate is around 5%, so there's a 95% chance that a male randomly selected from the general population has a job.  (Excuse me while I fix myself a stiff drink.)  We'll tackle her second and third requirements -- must not be married and must live nearby -- together.  From the year 2000 census statistics for Maris's current town (source: U.S. Census Bureau), I find that there are approximately 60,000 souls, of which roughly 4,500 are males between the ages of 25 and 34.  Unfortunately, obtaining information about marital status is a little tougher.  The percentage of married males between the ages of 20 and 34 is 33.6% (this being a national figure from the 2006 census), and the number jumps to 63.6% for males between the ages of 35 and 44.  These figures, along with basic intuition, suggest that more males get married as they age.  Consequently, we would expect the percentage of married males between the ages of 25 and 34 to be above the percentage for the larger 20 to 34 group.  So, let's crudely estimate that 40% of males between the ages of 25 and 34 are married, which means that 60% are unmarried.  That is, if Marisa restricts herself to unmarried, gainfully employed men between the ages of 25 and 34 in her town, the pool of suitable males is 4,500*0.95*0.60 = 2,565 dudes.  Now, things get a little murky.  Her man must have a strong dislike of the Packers, which to me seems tantamount to insisting that he hate beer, Santa, and breast implants as well.  Are there plenty of Yankees haters out there?  Sure.  The Red Sox?  Ten years ago, I would have said no, but today I'd have to say yes.  The Cowboys and the Patriots?  Plenty of people who don't like those franchises.  But, the Packers?  Good God, woman.  I imagine there are throngs of people who aren't fans of the Packers, but there's a much smaller number of people who actively and strongly dislike them.  Let's say that 75% of males are passionate enough about professional football to hate a particular team.  And, if we evenly distribute their hatred among the professional franchises, then let's say that roughly 3% of them hate the Packers.  Personally, I feel like this grossly overestimates the number of Packers haters, but onward we trudge.  With her Packers hatred requirement, Maris is now down to 2,565*0.75*0.03 = 58 guys.  But, wait, there's more.  Must be a Cubs fan.  Taking into account that Chicago is a major market and even if we include the hordes of bandwagon Cubs fans who couldn't pick Ernie Banks out of a bunch of emus, I can't imagine that the percentage of Cubbies faithful in a town 1,600 miles away is any more than 15% of the total population.  And, again, let's estimate that 75% of the male population is passionate enough about baseball to consider themselves a fan of a single team.  Maris has now whittled her list of suitable mates down to 58*0.75*0.15 = 7 guys.  In a town of approximately 60,000, Maris has a 0.01% chance that a randomly selected person from the population will be one of her suitable mates.  That's not 1%, people.  That's 1% of 1%.

Matters might actually be worse than that, however, because I've neglected all of the following things:  I've supposed that everyone is either truly single (that is, not dating anyone) or married.  So, unless Maris doesn't have a problem breaking up an engaged couple to get the guy, that makes things harder.  Moreover, we've made no mention of preferred education requirements or aesthetic desirability (the latter being the 800 pound gorilla that everyone's ignoring).  I've also assumed that Maris would date a person of any race, religion, political persuasion, income level, and sexual orientation.  Yikes.  Plus, I've supposed that she would date someone who doesn't speak English.  So, unless Maris doesn't mind dating a gay Mexican midget, we might need to start thinking about those magnificent seven dudes as an upper limit on the actual figure.

So, should you accept my calculated number as gospel?  Uh... probably not, because all of the dorks who are reading this are howling in disgust, and no fewer than three of them have probably had an aneurysm by now.  The big oversimplification I've made is in supposing that all of the preceding percentages are independent of each other.  For instance, I supposed that 75% of males are passionate football fans and 75% are passionate baseball fans.  But, presumably, there's some overlap there, and being a passionate football fan is probably correlated with being a passionate baseball fan, in which case simply multiplying 0.75*0.75 would not be valid.  Allow me to bore you to tears to further illustrate this point.

Suppose that HTV fanboy dap (hereafter referred to as HTVFD) has 3 red balls and 2 blue balls which he carries around in a large ballsack.  I know, I'm laughing too.  In the first experiment, we're going to consider "sampling with replacement" in which HTVFD is blindfolded by a scantilly-clad, buxom, chestnut-haired nymph so that he cannot see the contents of his ballsack. First, he selects a ball from his group of five balls, his lovely assistant records the color of the ball, and HTVFD places it back in the sack.  The impossibly beautiful assistant then jiggles HTVFD's ballsack to randomize the positions of the balls.  Next, he selects another ball from his ballsack, and again the nymph records the color of the ball.  Throughout this entire process, the buxom brunette coos and giggles and comments to HTVFD how big his biceps are, but this doesn't influence the outcome of the experiment.  What are the odds that HTVFD selects a red ball and a blue ball (without worrying about the order in which he selects them)?  Since each ball gets replaced after each selection, the probability of selecting a red ball during either step is 3/5, and the probability of selecting a blue ball during either step is 2/5.  Since the ball selections are independent (that is, the color of the first ball does not have any impact on the color of the second ball), we can state that the probability that HTVFD selects a red ball first and then a blue ball is the product of 3/5 and 2/5, or 6/25 = 0.24 = 24% and the probability that he selects a blue ball and then a red ball is also 24%.  Ergo, the total probability of selecting one red ball and one blue ball is 48%.

I realize nobody's still reading, but here I go anyway.  Let's now consider "sampling without replacement."  Here, we run through the same sequence of events as in the "sampling with replacement" scenario, with one important distinction:  after choosing the first ball from the ballsack, the beautiful assistant records the color, and then the ball is handed to the beautiful nymph's buxom younger sister, who has blonde hair and has just returned from her Pilates class.  She coos and giggles as well and remarks to HTVFD what powerful hands he has, though once again, I should caution the reader that this does not impact the outcome of the experiment.  HTVFD then draws a second ball from the ballsack and the experiment is complete.  Again, we ask the question: what is the probability that HTVFD selects a red ball and a blue ball, in either order?  The probability that he selects a red ball on the first try is again 3/5.  But, the probability of selecting a blue ball the second time around depends on what HTVFD selected first.  If he selected a red ball with the first trial, then there are 4 balls left in the sack, only two of which are blue, in which case the probability of selecting red then blue is the product of 3/5 and 2/4 or 6/20 = 30%.  But, he could also get a red ball and a blue ball by selecting a blue ball first (probability = 2/5) and a red ball second (probability = 3/4), with a probability of 0.4*0.75 = 30%.  Adding together the probabilities of selecting red-then-blue and blue-then-red, the total probability of choosing a red and a blue ball is 60%.  Granted, this might not seem to differ immensely from the 48% result in the "sampling with replacement" scenario, but the distinction can be important in many contexts.  In both scenarios, unfortunately, the probability that HTVFD will have sex with either the buxom blonde or brunette is 0.

Why am I boring the crap out of you with all of this?  It's a good question, and it occurs to me now that in a future post, I should really go on a 1,200 word tangent about some truly obscure math morsel completely unrelated to what I'm discussing, but not admit this until I've wasted your lunch hour.  Man, we'd all have a good laugh about that.  Anyhow, HTVFD's ballsack problem is relevant to Maris's inability to locate a suitable man, and I'm not going to make any joke here, because they're all too easy.  The ball selection problem was discussed to illustrate the point that if probabilities depend on other factors or events (as in the "sampling without replacement" scenario), you need to think harder about how you combine your probabilities.  You cannot simply multiply the "global" probabilities as I did in the calculation of suitable mates for Marisa.  

The question remains:  are there really only 7 Packers-hating, unmarried, employed male Cubs fans between the ages of 25 and 34 in Maris's town?  Probably not.  I don't know how much faith you can put in my calculation, but I would suppose it's not much.  Having said that, I've thrown a lot of numbers at you, and you're probably too exhausted to think about how you could potentially arrive at a better number, so you'll likely just accept my analysis as some reasonable approximation of the truth, even though I could be wrong by a factor of two, twenty, or fifty.  Which means that like any statistician worthy of the name, I've done my job.