Currently in Germany’s Top 40 Radio Charts is a song by Max Giesinger called „80 Millionen“, where the singer muses about the incredibly small chance of meeting his lover, a chance of one in eighty million, the rough-and-ready number of the German population. Though 81.8 million would be the more accurate number, for reasons of song-writing eighty million is fine. The singer also admits that he „was never good at probabilities“ but he remembered that the chance of meeting her (or him for that matter) approaches zero. Now the probability of meeting, falling in love and actually being together with one particular person is indeed close to zero – in the literal frequentist view it is one over eighty million, which is zero for all practical purposes, even for matters of the heart in which the brain has little say. The thing is, a purely frequentist approach is plain wrong. Luckily for Mr Giesinger, the chances are much better than he thinks.
First, assume that his song is about a woman, so we can safely say that the chance of meeting her just doubled to roughly one in forty million. (One in 41.5 million to be exact). Then, we know that Mr Giesinger was born in 1988, thus the creepiness rule of dating suggests that she is between 21 and 42 years old, and this is a conservative figure, the real figure for a young, handsome, fairly famous man like Mr Giesinger is probably much narrower, especially on the upper end, more like between 21 and 30. Germany’s statistical office puts the number of women in the age range of 21-42 in 2014 at 10.6 million, again raising the probability, this time four-fold. If we use the narrower, ad-hoc dating range of 21-30, we end up at 4.8 million possible partners for Mr Giesinger, implying a more than eight-fold increase in the probability as compared to the total female population.
Until here it is all about frequencies, and every woman in the 10.6 or 4.8 million possible candidates has an equal probability, so the song should be more accurately named „10,6 Millionen“ or „4,8 Millionen“, depending on Mr Giesinger’s age preferences. But that’s not all. Usually you meet your future partner in something you have in common, such as university/school, work, hobbies, etc. After all, you have to actually meet your partner sooner or later, and the probability of meeting someone with similar hobbies is larger. The same is true, if she attends the same university programme – or, especially in smaller towns, only lives a few streets away from you. Now, Mr Giesinger lives in Germany’s second city Hamburg with a population of 1.75 million, still the number of suitable partners is rather small, at 292753 (21-42) or 134563 (21-30) Hamburgian women in the right age cohort.
But this could be narrowed down even further, to women working in his industry or visiting the clubs, where he plays. Unfortunately (luckily, actually!), there are no data on that. So let’s stick at first with a number in the range of some hundred thousand. My guess would be that this number can be narrowed down two additional orders of magnitudes to a few thousand women. Now, Mr Giesinger’s musical estimate of the chances are one in eighty million, which relates to a probability of 8×10⁻⁷, whereby the more realistic estimate is arguably more like 10⁻⁵, that is, an increase of probability by a factor of one hundred. Mr Giesinger has a chance of meeting her that is a hundred times higher, than he originally estimated. And this is only for the conservative number of possible women in Hamburg. Using the ad-hoc number of a few thousand, which I find more realistic, the probability of meeting her is rather 10⁻³, i.e. ten thousand times likelier than he originally estimated.
OK, Now what?
Please view this little exercise tongue-in-cheek. The song is quite likeable, actually one of the better (German-language) songs currently on air, which doesn’t really count for much, but I digress. So while this is a fun exercise illustrating the matching problem and how far off the probability estimation of the guy-next-door can be, unfortunately such statistical illiteracy incur costs on people themselves and on the economy as a whole, when the general public gets the probabilities of events wrong. Let me give you a few examples: The probability of living a longer life increased in the recent decades, still our pension system is tweaked Bismarckian and the old-age savings of Germans are far too small. Second, the probability of a catastrophic nuclear meltdown is low, still Germany rids itself of a powerful means against climate change, the probability of which is one. The probability of being killed in a car accident is orders of magnitude higher than the probability of a plane crash, still people race like crazy on the Autobahnen instead of taking a plane due to Flugangst. There are more examples of when people do not get their probabilities right, so statistical illiteracy or „not being good at probabilities“ as Mr Giesinger put it, is not only a shame for a rich country like Germany but also costly. Still, if Mr Giesinger were better at probabilities, he arguably would not have become a singer, and in turn not providing me the opportunity for another whimsical blog post.
(This actually reminds me of another whimsical treatment of the song „Nine Million Bicycles“ by Katie Melua a few years ago. )