Why do these probabilities not sum to 1? The problem involves me owning 2 black cows (B), 3 white cows (W), and 1 mixed cow (M) that is black on one side and white on the other.
I look out my window and see two cows standing with black sides facing me - the event bb.
The probability of the cows being one black and one mixed - the event MB - is:
---- note: P(BB) = (2C2)/(6C2) = 1/15, and P(MB) = (1C1)(2C1)/(6C2) = 2/15 ----
P(MB|bb) = P(bb|MB)P(MB)/[P(bb|MB)P(MB) + P(bb|BB)P(BB)]
P(MB|bb) = (1/2)(2/15)/[(1/2)(2/15) + (1)(1/15)] = 1/2
The only other possibility, when bb is given, is BB (that the two cows are black), but:
P(BB|bb) = P(bb|BB)P(BB)/[P(bb|BB)P(BB) + P(bb|MB)P(MB)]
P(BB|bb) = (1)(1/15)/[(1)(1/15) + (1/2)(2/15)] = 1/4
What event is related to the remaining 1/4?
EDIT: They don't sum to 1 because I can't count. Thanks for the help guys!

There are 8 ways to get the cow to be black. 4 ways to get the two black cows in any of the sides. 
The other 4 ways are determined by getting the mixed cow in the right position picking which of the black cows gets in and which of the black sides of the black cow is shown.
