Wednesday, November 16, 2005

The paradox of the unexpected fire drill

A message came round in mid-October, saying that we were going to have a surprise fire alarm practice some time in November. All we were told is that it would happen at 3pm one day, but we would not know which day it was going to be until the alarm went off.

So I got to thinking about what day was most likely. It was immediately obvious that it can't be scheduled for the 30th, because if it has not happened before we come into work on that day, we would know it has to happen that afternoon, and therefore it would not be a surprise. But on the morning of the 29th, there would only be two possible days left, and having already ruled out the 30th as impossible, it would have to be the 29th. So there is no surprise there either. Having ruled out the 30th and 29th, it is easy to show that the 28th is similarly impossible. And so on, right back to the 1st. I conclude that we can't have the surprise fire drill at all!

I know you will all be fascinated to hear how things turn out.

Update 16 Nov

Well, it happened today. Was it a surprise? No not really, as someone had already told me it was planned for this day. In fact I made up the story about it being a surprise. But it was only after glancing at the Wikipedia page on the paradox of the unexpected hanging that I found out that this is also known as the paradox of the unexpected fire drill. If you still don't know what I'm talking about then read the page and all will become clear.

6 comments:

CapitalistImperialistPig said...

My guess is that if they really wanted to surprise you, they wouldn't have told you in advance. Administrators don't like surprizes because they can make them look bad. Count on it being on the twenty-eighth.

Anonymous said...

Somehow I'm unable to shake the image of all those OL's racing outside in their fur-lined lingerie. Please don't ruin it for me by describing what it really looked like.

EliRabett said...

I understand this is how executions are carried out in Japan

James Annan said...

Chris,

I think you are overcomplicating things a bit. I don't think you have to go as far as wondering about methods etc.

Consider the simpler version: "The fire drill will be at 2pm tomorrow, and it will be a surprise". There is no decision to make. It seems to me that it is closely related to the simple "this statement is false" paradoxes - or maybe the "this statement is unprovable" of Godel's incompleteness theorem. In fact, if you swap "provable" for "believable" then it seems to be an near-exact analogue. Consider "this statement is not believable". It is true, but cannot be believed to be true. The statement "you will be hanged tomorrow and will not expect it" is also true but cannot be believed.

James Annan said...

Chris,

I'm not convinced. The paradox is unchanged even when only one day remains - the sentence is still true but not believable. I also note that even the somewhat simpler "liar paradox" ("this sentence is false") seems to have generated a fair amount of philosophical debate which I am not qualified (or sufficiently motivated!) to wade through. Given that the English language contains simple syntactically-valid sentences which can be neither true nor false, it seems to me that the existence of a sentence which is true but not believable is not really such a big deal.

Anonymous said...

"The fire drill will be at 2pm tomorrow, and it will be a surprise" is indeed not believable, but it may be either true or false. The same goes for the judge's statement. The prisoner reasons that he can't be hung on Friday, else the judge's statement would be false. But that's ridiculous -- indeed the judge's statement *could* be false, so the prisoner *can* be hung on Friday. And since he can be hung on Friday, there's certainly no reason why he can't be hung on Thursday, Wednesday, etc. Certainly a prisoner who is consigned to his fate but uncertain of the day of hanging won't be *surprised* when the hangman shows up, regardless of when that is. Only the prisoner who has bogusly reasoned that he can't be hung will be surprised -- but the judge can't be certain that the prisoner has so reasoned. The judge is just making utterances, possibly true and possibly not, and such utterances have no bearing on what is possible and what is not.