2018-10-09
The combinatorics of matchingSaw a YouTube video with a professional astrologer. He was given 12 subjects (one from each zodiac sign) and asked to determine which was which. He ended up getting 4 of the 12 right. The guy had phenomenal cold reading skills, which he demonstrated both before and after learning the subjects' actual signs. But anyway it got me wondering about the probability of his getting such a result by chance.
This was a new area of combinatorics for me. The number of ways to assign exactly 4 to the right pigeonholes is the number of ways to get exactly 8 all in the wrong pigeonholes. The number of ways to pick the 4 (or 8) can be found with the familiar "x choose y" formula, but having done so, how many arrangements of the 'unlucky 8' successfully avoid putting any one of them where it belongs? I was not familiar with any combinatorical methods that addressed a special correspondence between the objects permuted and the positions they are permuted into.
I figured it out. Chase the link if you want the details, but the short answer is 1.9%. If you assign 12 zodiac signs at random to people from all 12 signs many times, you would get four or more correct 1.9% of the time.
That used to be good enough to publish a scientific paper, but the replication crisis has drawn people's attention to the fact that 'happens by chance multiple times in a hundred trials' is a weak standard of evidence that Fisher pulled out of thin air in the first place, so I hope it would get some side-eye now. I mean, how many statistical significance tests get published every day?
And, of course, that presumes that the participants' personae, their ways of presenting themselves that served as cues to the astrologer, were not in any way (including subconsciously) influenced by a knowledge of their sign and of the traits typically associated with each. If you're playing to type to begin with, that makes this less impressive. I don't know how the subjects were selected. If it was walking into a workroom and saying "Hey, who wants to meet an astrologer?" this doesn't mean anything.
Now, to find out if the function for matching n of m objects to unique 'home' slots has a name. I checked earlier for an integer sequence that matched it, but my numbers were wrong at the time, so of course I didn't find one. Edit: They're called partial derangements or rencontre numbers.
20:28
