# 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 egg from 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 thought I 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.