Re: help with math problem
 

A very accessible page on bayes' theorem:
http://yudkowsky.net/bayes/bayes.html

Re: help with math problem
 

Your answer is fine (although your first two probabilities should both be divided by 2 since you start with a 50% chance of picking either bucket first). The key feature of the problem is that you're 28 times more likely to have initially picked the 8B2R bucket, and you come up with that for the right reason.

Re: help with math problem
 

More simply put:

The chances you picked the first hat and got two blues is 1/2 x 2/10 x 1/9 or 2/180.

The chances you picked the second hat and got two blues is 1/2 x 8/10 x 7/9 or 56/180.

You are going to get two blues 58 out of 180 times. When you do, there is a 56 out of 58 chance it came from the second hat.

So there is a 56/58 chance that you are looking at an 8/10 probability (to pick a red) and a 2/58 chance you are looking at a 2/10 probability.

56/58 x 8/10 plus 2/58 x 2/10 is 452/580.