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 World According to Physics - Jim Al-Khalili *****

There is a temptation on seeing this book to think it's another one of those physics titles that is thin on content, so they put it in an odd format small hardback and hope to win over those who don't usually buy science books. But that couldn't be further from the truth. In Jim Al-Khalili's The World According to Physics, we've got the best beginners' overview of what physics is all about that I've ever had the pleasure to read.

The language is straightforward and approachable. Rather than take the more common historical approach that builds up physics the way it was discovered, Al-Khalili starts with the 'three pillars' of physics: relativity, quantum theory and thermodynamics. In simple language with never an equation nor even a diagram in sight, the book lays out what physics is all about, what it has achieved and what it still needs to do.

That bit about no diagrams is an important indicator of how approachable the text is. Personally, I'm no…

Until the End of Time: Brian Greene ***

Things start well with this latest title from Brian Greene: after a bit of introductory woffle we get into an interesting introduction to entropy. As always with Greene's writing, this is readable, chatty and full of little side facts and stories. Unfortunately, for me, the book then suffers something of an increase in entropy itself as on the whole it then veers more into philosophy and the soft sciences than Greene's usual physics and cosmology.

So, we get chapters on consciousness, language, belief and religion, instinct and creativity, duration and impermanence, the ends of time and, most cringe-making as a title, 'the nobility of being'. Unlike the dazzling scientific presentation I expect, this mostly comes across as fairly shallow amateur philosophising.

Of course it's perfectly possible to write good science books on, say, consciousness or language - but though Greene touches on the science, there far too much that's more hand-waving. And good though he i…

Jim Al-Khalili - Four Way Interview

Jim Al-Khalili hosts The Life Scientific on BBC Radio 4 and has presented numerous BBC television documentaries. He is Professor of Theoretical Physics and Chair in the Public Engagement in Science at the University of Surrey, a New York Times bestselling author, and a fellow of the Royal Society. He is the author of numerous books, including Quantum: A Guide for the Perplexed; The House of Wisdom: How Arabic Science Saved Ancient Knowledge and Gave Us the Renaissance; and Life on the Edge: The Coming of Age of Quantum Biology. The paperback of his novel Sunfall is published in March 2020 by Transworld. His latest book is The World According to Physics.


Why physics?

I fell in love with physics when I was 13 or 14, when I realised not only that I was pretty good at it at school – basically common sense and puzzle solving – but because it was the subject that answered the big questions I had started contemplating, like whether the stars in the night sky went on for ever, what they were ma…