> If you look at the Bayes probabilities for the 3 cakes
>
> P(BB)=1/3 prob to get Brown/Brown cake
> P( not BB)=2/3 prob of not BB cake
>
> P(side B/BB)=1 you are looking at Brown side
> P(side B/not BB)=1 you are looking at Brown side

Why? There are two pancakes GG and GB with four sides, so that

P(side B/not BB) = 1/4.

If you substitute that into your Bayes formula

> P(BB/side B)=P(side B/BB)* P(BB) / ( P(side
>B/BB)*P(BB)+P(side B/not BB)*P( not BB) )

you get

P(BB/side B) = 1·1/3 / (1·1/3 + 1/4·2/3)
  = 1/3 / (1/3 + 1/6)
  = 1/3 / (3/6)
  = 6/9
  = 2/3