Seeing Bayes' Theorem

I'm a journalist, so I'm contractually obliged to be innumerate. I can't therefore understand Bayes' theorem - I get me Pr(A¦B) and Pr (B¦A) mixed up. Sums is hard. So I use pictures instead.
To see how, take this from Glen Whitman.

Suppose that stupid people constitute 60% of the population (a generously low number). Suppose that smart people split 50-50 between conservatism and liberalism. And suppose that stupid people split 80-20 between conservatism and liberalism, consistent with Mill’s claim. Now, say you meet a random person who turns out to be conservative, and you know nothing else about him. How likely is he to be stupid?

Being a smart fella, he uses Bayes' theorem. Being stupid, I use boxes, like below. First, I fill in the bottom row;  out of 100 folk, 60 are stupid and 40 smart. As the 40 smarties split equally between Conservatives and Liberals, I can put 20 into each box in the smart column. This leaves only the "stupid" column to fill in. As the 60 stupids split 80-20 for the Conservatives, I figure this means they split 48-12 - hey, I can do percentages. 
Having filled in all the boxes, I can answer Glen's question. There are 68 Conservatives, 48 of whom are stupid. So the chances of a random Conservative being stupid are 48 out of 68. Which is just under 71%.
Exactly what Glen says. Without any sums harder than simple percentages.
This Bayes malarkey ain't so hard, is it?

Comments

[zorro]

I'm sorry are you saying

Conservative != Liberal

? I don't think these two words mean opposite things, it is quite possible to be a conservative and have liberal view. Or is this post purely aimed at Americans where "liberal" = "Democrat"?

In this country at least you would have to be barking insane to vote for the 'liberal' democrats, who would undoubtedly give even more of our hard earned money to our new Euro-Godfathers.

And if you are trying to use this 'theory' as proof that conservatives are more stupid then I suggest you re-examine your hypothesis.

Note "And suppose that stupid people split 80-20 between conservatism and liberalism, consistent with Mill’s claim. "

Well suppose the stupid people are split 10-90 between Conservatives and Labour supporters. A MUCH more realistic figure in this country at this time.

[chris]

The Conservative-Liberal split and the numbers are unimportant. It's about Bayes theorem. That's all.

[mat (in a slightly tongue-in-cheek mood)]

Thank you for this proof, Chris. I can now see that Mill was right. Just one thing... in modern Britain, the Labour Party is the relatively conservative party, and the Conservative Party is the relatively liberal party.

And, I know you said the numbers are unimportant, so I will only go so far as to venture that smart people split 0 to 100 between conservatism and liberalism. Otherwise they wouldn't be smart, would they?

Assuming, of course, we're talking about classical liberalism / libertarianism. The philosophy of conservatism in the UK presently implies sticking with authoritarianism / socialism.

[Mark Wadsworth]

It must be said that some mathematicians and statisticians deliberately mystify the whole thing. I'd have done boxes as well (at least in my head) and come to the same conclusion.

If it's more tricky than that I use a spreadsheet, e.g. if you have 2,000 songs on your mp3player and listen to music for two hours a day (=30 songs) on a truly random setting, what it the chance that on any day you hear the same song twice?

Answer appears to be one-in-five.

[dearieme]

To see your point made very clear indeed, with lovely examples, read Gerd Gigerenzer's "Risk" (Penguin 2003). One of the best paperbacks I've read in ages (which is why I recommend it here from time to time).

[Glen Whitman]

You're right, that's another way of doing it. In fact, that's what I did to make sure my calculations were correct before making that post. The advantage of Bayes' Rule is that you can find a general formula like the one at the end of my post. Also, the tabular approach can get confusing when you're dealing with very small probabilities, such as with disease testing.

[Karthik]

Thank you for the proof.
I have picked up a tool, or should I say method, to use in the future.
Thank you very much.

[james c]

Does it really take all of this to calculate that there are 48 stupid conservatives, 20 smart ones and so the chance of a a random conservaive being stupid is 48/(48+20)?

By the way, the reason people argue with your Mill/stupid/conservative/post is not because they can't add up, but because you use the word stupid to mean different things. Mill uses it, in one place, to mean 99% of the population. Most readers will take it to mean a much smaller proportion of the population.

[ManBearPig]

To see your point made very clear indeed, with lovely examples, read Gerd Gigerenzer's "Risk" (Penguin 2003).