What is probability?

I haven't seen any particularly interesting (or unusually egregious) uses of statistics in the last few days, so it's time to have a more philosophical outing.  Of course, if you see anything in the news you think is worthy of a post, email me (evans [at] stats [dot] ox [dot] ac [dot] uk).

So to the title of the post - what is probability?  Probability can be viewed as just a mathematical construct, which we'll explore some other time, but even with the mathematical rules in place, there are various interpretations of what a probability actually is.  At school we usually think about rolling dice and flipping coins, so let's start there.  I flip a 50p coin: what's the probability that it comes up heads?  You'd probably [sorry] say it's 'evens', 'a half', 'fifty-fifty', or (as mathematicians generally prefer), '0.5'.