# Two barrels problem from the Intuitive Explanation (answered)

I’m not sure if I’m doing something wrong here. EDIT: Yup, I’m allowing myself to be tricked.

I’ve finally sat down to reading http://yudkowsky.net/rational/bayes carefully, and I solved all story problems so far with no trouble. However, now I’m at this one:

Q.Suppose that there are two barrels, each containing a number of plastic eggs. In both barrels, some eggs are painted blue and the rest are painted red. In the first barrel, 90% of the eggs contain pearls and 20% of the pearl eggs are painted blue. In the second barrel, 45% of the eggs contain pearls and 60% of the empty eggs are painted red. Would you rather have a blue pearl egg from the first or second barrel?

A. Actually, it doesn’t matter which barrel you choose! Can you see why?

This doesn’t look right to me.

In the first barrel, we have 18% blue eggs that contain pearls, and an unknown number of blue eggs that do not contain pearls, anywhere between 10% (worst case) and 0%. Depending on that, the proportion of blue eggs with pearls among all blue eggs can only be between 18/(18+10) = 64ish% in the worst case, to 100% in the best.

In the second barrel, we don’t know how many pearls eggs are blue. We do know there are 45% eggs with pearls altogether, therefore 55% without pearls, and out of the latter 60% are red therefore 40% are blue. That means we have 40%*55% = 22% empty blue eggs. Pearl blue eggs are anywhere between 0 and 45%, so from 0% to 45/(45+22) = 67ish%.

Were we just supposed to conclude that there isn’t enough information to answer that problem? But I’d say “anywhere between 64% and 100%” is a better shot than “anywhere between 0% and 67%”. If I actually had to choose, and there were valuable pearls at stake, I’d choose the first barrel. Am I making some sort of a mistake?

This is like the question, “Which is heavier, a pound of feathers or a pound of gold?”

The question is, would you rather have

a blue pearl eggfrom the first or second barrel?You know that this hypothetical egg is blue. And has a pearl in it. It doesn’t matter which barrel the egg is from because these are the only dimensions along which eggs vary.

Hahaha, yeah. Well that was slightly embarassing. My brain just didn’t bother parsing the actual question, just assumed I’m being asked about blue eggs again.

What’s scary is that I

thoughtI carefully re-read the problem after I noticed I’m unsure I understand it, and I still didn’t see it; I just paid even more attention to the %s.The traditional even-more-stupidly-clever answer to that is that a pound of feathers weighs more, because precious metals are measured in troy pounds rather than avoirdupois pounds.

Nice. If I were to try to give one like that to the barrels problem, I’d say “barrel 2 please”, because that way, if afterwards I’m asked to choose a barrel to get a random blue egg after all, I won’t have shot myself in the foot by lowering the favorable odds from barrel 1..

This reminds me of those tests in grade school that tell you to read all the instructions before starting, then to do all kinds of silly stuff, then to ignore previous instructions. I wonder if careful reading is good for things other than not getting tricked by such jokes. (And accidental instantiations thereof on written tests.)

There are a few other cases I can think of offhand where careful reading is important. Decoding poorly written instructions and this card game in particular came to mind immediately, but on reflection I think the best example in my experience would be coding, and in particular debugging. I’m by no means an expert (or necessarily a representative sample), but at least half of the bugs I spend 10+ minutes chasing around end up being caused by a minor brain-dead error in a single expression.

Even if your interpretation were correct, the conclusion “not enough information” wouldn’t. You have to decide somehow no matter how little information is available. However there may not be a clearly unique “correct” answer when there is too little information, as more priors may be reasonable.