Concidence, probability and asking the right question

Our chief consultant writes:

We’ve all had it happen: some unexpected, unusual occurrence, and someone asks: “What were the odds against that happening?” As an exclamation, an alternative to the pedestrian, “That’s unusual,” this is fine. Language should have flexibility and the freedom of metaphor. This becomes a problem, however, when the unusual event is taken to imply unknown laws of physics or perhaps sinister forces at work.

Actually working out probabilities in any but the simplest cases can be pretty tedious, and we’re not about to get into that here. But it’s easy to make basic mistakes in setting up this kind of question (respected scientists have done so); we present two rules to help keep you out of trouble, even if you’re not going to punch any numbers into your computer.

When you want to figure the odds of something happening, be sure

• You ask the question without proper nouns, and
• You ask it before you know the outcome.

I’ll give you an example to illustrate how to violate the rules. A professor of astronomy at a prestigious institution we shall not name wrote a book for the public. In it, he asserted that the multiple star Theta Orionis must be associated with the diffuse glow of the Orion Nebula, rather than appearing by chance in the same direction in the sky. Since the Nebula is about a square degree in area, and there are forty thousand (roughly) square degrees in the sky, the probability of a chance alignment is one in forty thousand—surely negligible.

What is the actual probability of Theta aligning with the Nebula? One. There is exactly one Theta Orionis in the universe, and that’s where it is.

How should this have been done? Decide beforehand what you will consider to be a significant alignment. Getting away from proper nouns, pick which stars are interesting (quadruple stars like Theta, of a given brightness or brighter) and what nebulae are to be considered (diffuse nebulae brighter than a certain cutoff). Decide how close the alignment has to be. Then go to the sky and see what you get. You’ll get a probability that any interesting star will align with a nebula, from which you can later decide how unusual Theta is.

Now, Theta Orionis is indeed intimately connected to the Orion Nebula; we know that from much other data. And the book was written well over a century ago, with the unnamed prestigious institution having done a great deal of good statistical work since. So this is not a current or important blot on the science of astronomy. It does, however, show how to get things wrong.

Let’s come back to the mundane. Suppose you’re at a baseball game, and a famous batter hits a line drive that lands somewhere on the grass. Your friend next to you says, “Wow! What’s the chance that that ball would land on that exact blade of grass for his 100th hit of the season?”

To ask the question correctly we have to start before the hit and leave out proper nouns (“that ball . . . that exact”). It certainly isn’t unusual for a line drive to hit grass (depending on the skill of the fielding team); the probabilities only get small if you can choose your blade of grass beforehand and manage to get it hit.

If you ask the question the wrong way around, you can be startled by something quite ordinary occurring or wind up being certain that something highly unlikely will happen.

These may seem subtle points, more suited to philosophers or scientists than people in their normal lives. I certainly don’t intend to set you a series of exercises in probability to test you. But it’s common to find someone arguing that an event is highly unusual and needs to be explained. You might check this sort of thing next time it comes your way.

(The matter of Theta Orionis, as well as a lot of other stuff, appears in our astronomer’s book Hindsight and Popular Astronomy.)