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.
And yet I still buy a lottery ticket. I think I enjoy the hope... no matter how fleeting rather than probability (no matter how infinitesimal).
ReplyDeleteYou're right of course, Steve. Despite the logic of the situation, I buy a lottery ticket from time to time. It's clear that what I'm buying is the right to imagine what I'd do with the prize money.
ReplyDeleteBut it also serves as a warning: if I buy a ticket too often, I should look at what I need to fix in my life.
That was a super explanation which did not make my brain hurt.
ReplyDeleteAnd I've never bought a lottery ticket - but used to back the geegees.
Thanks, Helen! I have never been to a race track to bet, though I have shouted at televisions during the Grand National.
ReplyDeleteThings happen that have no chance of happening. Which is why predicting the future is a dodgy business.
ReplyDeleteOr an infinitesimal chance?
Hi James, the problems arise when we try to translate perfect mathematical models into physical reality. My discussion blithely assumes that it is possible to generate a random number between 0 and 1 with arbitrary precision.
ReplyDeleteIn the physical example using the pointer, I am assuming, apart from the concept of a perfect pivot, that when the pointer stops, we know exactly where it is. Heisenberg tells us that we can't know its momentum and position both at the same time, so there is always going to be some uncertainty as to its position.
I suspect that, outside of mathematical perfection, probability is quantised, and therefore, when the physical probability is zero, things truly never happen, and otherwise there is a minimum, infinitesimal probability that they do happen.
I find it interesting that a simple fairground example can lead us so easily into this kind of deep physical problem.
(The uncertainty in the position of the pointer means that we can only say that it lies within an interval, so our calculations yield a non-zero probability)
ReplyDelete