## Wednesday, June 9, 2010

### probability zero is not impossible, or is it

Last night at the ol' quizzor, the question arose:

Suppose identical twins marry identical twins. Will their children be identical?

the answer they were looking for was 'no'. The answer I put down was 'maybe, but it is very unlikely'. We were marked wrong - but given a bonus beer (small consolation at the time, but finishing 3 points clear at the end meant it was an ideal trade). The quizmaster's defense? 'it is no more likely than you being identical to your brother!'

In fact, it's half as likely (given that I'm male and so is my brother we've got a little extra chance built into the formulation), but both are still possible. Yes, they're unlikely: but the probability is positive.

That led me immediately to thoughts of the difference between 'probability zero' and 'impossible'. Despite what you were told in school they are not one and the same.

Choose an integer at random. Assuming you can do such a thing (not a very reasonable assumption....but whatever), the probability for a specific integer is zero. Not 'very small', but zero. So every event has probability zero - and so whatever you chose had probability zero. It was impossible, and yet it happened. How miraculous.

There are plenty of cases in mathematics where this comes up, but very few in the real (ultimately finite) world. However, too many people take this real-world intuition and drag it, kicking and screaming, into the abstract world and treat it as fact.