Skip to main content

Fun with the Reverend Bayes

A recent review of Bayes' Rule by James V. Stone for review, has reminded me of the delightful case of the mathematician's coloured balls. (Mathematicians often have cases of coloured balls. Don't ask me why.)

This is a thought experiment that helps illustrate why we have problems dealing with uncertainty and probability.

Imagine I've got a jar with 50 white balls and 50 black balls in it. I take out a ball but don't look at it. What's the chance that this ball is black?

I hope you said 50% or 50:50 or 1/2 or 0.5 - all ways of saying that it has equal chances of being either white or black. With no further information that's the only sensible assumption.

Now keep that ball to one side, still not looking at it. You pull out another ball and you do look at this one. (Mathematicians know how to have a good time.) It's white.

Now what's the chance that the first ball was black?

You might be very sensibly drawn to suggest that it's still 50:50. After all, how could the probability change just because I took another ball out afterwards? But the branch of probability and statistics known as Bayesian tells us that probabilities are not set in stone or absolute - they are only as good as the information we have, and gaining extra information can change the probability.

Initially you had no information about the balls other than that there were 50 of each colour in the pot. Now, however, you also know that a ball drawn from the remainder was white. If that first ball had been black, you would be slightly more likely to draw a white ball next time. So drawing a white makes it's slightly more likely that the first ball was black than it was white - you've got extra information. Not a lot of information, it's true. Yet it does shift the probability, even though the information comes in after the first ball was drawn.

If you find that hard to believe, imagine taking the example to the extreme. I've got a similar pot with just two balls in, one black, one white. I draw one out but don't look at it. What's the chance that this ball is black? Again it's 50%. Now lets take another ball out of the pot and look at. It's white. Do you still think that looking at another ball doesn't change the chances of the other ball being black? If so let's place a bet - because I now know that the other ball is definitely black.

So even though it appears that there's a 0.5 chance of the ball being black initially, what is really the case is that 0.5 is our best bet given the information we had. It's not an absolute fact, it's our best guess given what we know. In reality the ball was either definitely white or definitely black, not it some quantum indeterminate state. But we didn't know which it was, so that 0.5 gave us a best guess.

One final example to show how information can change apparently fixed probabilities.

We'll go back to the first example to show another way that information can change probability. Again I've got a pot, then with 50 black and 50 white balls. I draw one out. What's the probability it's black? You very reasonably say 50%.  So far this is exactly the same situation as the first time round.

I, however, have extra information. I now share that information with you - and you change your mind and say that the probability is 100% black, even though nothing has changed about the actual pot or ball drawn. Why? Because I have told you that all the balls at the bottom of the pot are white and all the balls at the top are black. My extra information changes the probabilities.

Comments

Popular posts from this blog

The Feed (SF) - Nick Clark Windo ****

Ever since The War of the Worlds, the post-apocalyptic disaster novel has been a firm fixture in the Science Fiction universe. What's more, such books are often among the few SF titles that are shown any interest by the literati, probably because many future disaster novels feature very little science. With a few exceptions, though (I'm thinking, for instance, The Chrysalids) they can make for pretty miserable reading unless you enjoy a diet of page after page of literary agonising.

The Feed is a real mixture. Large chunks of it are exactly that - page after page of self-examining misery with an occasional bit of action thrown in. But, there are parts where the writing really comes alive and shows its quality. This happens when we get the references back to pre-disaster, when we discover the Feed, which takes The Circle's premise to a whole new level with a mega-connected society where all human interaction is through directly-wired connections… until the whole thing fails …

The Bastard Legion (SF) - Gavin Smith *****

Science fiction has a long tradition of 'military in space' themes - and usually these books are uninspiring at best and verging on fascist at worst. From a serious SF viewpoint, it seemed that Joe Haldeman's magnificent The Forever War made the likes of Starship Troopers a mocked thing of the past, but sadly Hollywood seems to have rebooted the concept and we now see a lot of military SF on the shelves.

The bad news is that The Bastard Legion could not be classified as anything else - but the good news is that, just as Buffy the Vampire Slayer subverted the vampire genre, The Bastard Legion has so many twists on a straightforward 'marines in space' title that it does a brilliant job of subversion too.

The basic scenario is instantly different. Miska is heading up a mercenary legion, except they're all hardened criminals on a stolen prison ship, taking part because she has stolen the ship and fitted them all with explosive collars. Oh, and helping her train her &…

On the Moor - Richard Carter ****

There's much to enjoy in Richard Carter's pean to the frugal yet visceral delights of being one with England's Pennine moorland. If this were all there were to the book it would have made a good nature read, but Carter cleverly weaves in science at every opportunity, whether it's inspired by direct observations of birds and animals and plants - I confess I was ignorant of the peregrine falcon's 200 mile per hour dive - or spinning off from a trig point onto the geometric methods of surveying through history all the way up to GPS.

Carter is something of an expert on Darwin, and inevitably the great man comes into the story many times - yet his appearance never seems forced. It's hard to spend your time in a natural environment like this and not have Darwin repeatedly brought to mind.

I confess to a distinct love of these moors. Having spent my first 11 years in and around Littleborough, just the other side of Blackstone Edge from Carter's moor, the moorland…