I happened to be listening to Sting's Ten Summoners' Tales recently, my favourite track from which is "The Shape of my Heart". It discusses the hidden worlds of probability and chance that underpin our daily lives, and I can never hear it without musing on the strange fact that things happen that have no chance of happening.
Mathematical probability discusses the chances of certain events happening at some time in the future, and since it's therefore essentially about fortune telling, perhaps it's not surprising that it's sometimes treated distainfully. "Something either happens or it doesn't", "probability is irrelevant", and so on. But it has useful applications, such as telling a company how little it can charge you for life insurance whilst there being a good chance of it still being in business should it have to pay out.
Mathematicians assign a probability between zero (no chance) and one (certainty) that an event will happen under specified circumstances. And when an event has zero probability, they say it "almost never happens". Why "almost never" rather than "never"? They're hedging their bets.
If we imagine a line of length 1, the points along it represent all possible numbers between 0 and 1. If we then take a random number between 0 and 1, this will correspond to a point somewhere along this line. The probability our random number lies between 0 and 0.5 (halfway along the line) is 0.5, as is the probability that our number lies between 0.5 and 1. In general, the probability that our random number lies in the interval between two points A and B that lie on the line (with A less than B) is B-A, i.e. the length of the line from A to B.
We can make the distance between A and B as small as we like, and ultimately, when B=A we don't have a length of line, we just have a point. The probability that our random number is exactly equal to A is then A-A or zero. (If you like, there is an infinity of points along a line of length 1, so the probability that our random number is equal to a specific point, is one in infinity, or zero)
So now we take our line of length 1, bend it into a circle, and pivot a spinning pointer in the middle. When our pointer comes to rest, it will point at a specific number. The probability, when we spun the pointer, that it stopped exactly at this number, was zero, but none the less it stopped there. So an event with zero probability has actually happened.
Things happen that have no chance of happening. Which is why predicting the future is a dodgy business.
As a corollary, in practical terms, you have no chance of winning the lottery, even though someone wins it most weeks.