When improbable events are expected

This week’s column by The Numbers Guy in the Wall Street Journal focuses on improbable events. Several events that seem intrinsically unlikely have happened in recent weeks: two nuclear submarines collided in the vastness of the oceans; two satellites collided in the vastness of space; and the two engines of a 737 airliner were disabled simultaneously by birdstrike. This last one is probably the least surprising of the three: there are several thousand birstrikes on civil aircraft every year, some of which lead to engine failure. Given that birds travel in flocks, it’s not too surprising that two engine failures might happen simultaneously (these are hardly independent events!).

The other two seem, at first blush at least, seem more surprising. Space is big … really big. So are the oceans. And there are relatively few satellites and submarines occupying them. It’s true to say that in any given minute the chance of a collision between any two submarines is small, assuming they’re wandering around the ocean randomly. Wait enough minutes, and narrow their range from the entire ocean to standard patrol routes, and what at first seems astronomically impossible eventually becomes an actual collision. It’s the real-world manifestation of the Birthday Paradox: if I pick two random strangers off the street, there’s less than a 0.3% (1/365) chance they share the same birthday. But put 23 random strangers in the same room, and there’s better than a 50% chance that two of them will be eating birthday cake on the same day.

In a follow-up post detailing the calculations from the WSJ article there’s another example from statistician Jessica Utts, author of “Seeing Through Statistics.” I haven’t read that book, but the quotation makes me think it will be interesting:

… Events that seem unlikely at first glance may not be that unlikely given enough opportunity. For example, Utts says, if everyone dreamed about a plane crash once in their lives, a few thousand people would have the crash dream on any given night — including the night of a crash. “These specific incidences may be unlikely, but the combined probability of something similar at some point in time is probably fairly high.”

But it’s this quote from statistician Peter Westfall that sums up the discussion perfectly for me: