The problem I have is that you're assuming that each of the 27 outcomes has equal probability, but the chance that we received the information in the way we did makes that a flawed assumption.

The best analogous problem is the German Tank Problem:

http://en.wikipedia.org/wiki/German_tank_problem

If we have destroyed a single German tank with a serial number 100, we can at least to begin to make an estimate on the size of the German force, by basically asking the question:

"If they have 200 tanks, what was the chance one we randomly killed was this serial number? 500? 1000?"

And then combining n=100->infinity to form a probability distribution. You can then say that there is an x% chance that Germany has 500 tanks, and a y% chance that Germany has 10,000 tanks.

However - if instead, we asked 'does there exist a German tank with a serial number 100', and the answer is yes, this does NOT tell us anything past the fact that their tanks are >= 100 in number.

We have the exact same information, but how it was determined changes the outcome drastically.

Does that make sense?