So I was asleep for this whole discussion, but I just finished reading through it now. I meant to say in my post last night that there might be a way of phrasing the problem to get Haddock's answer, and as I was thinking about it this morning, I realized that there was, so I semi-take back what I said in the other thread, because the problems I addressed there were ambiguous (my solutions were correct only for a particular understanding of the problems).
As for the actual game, I am pretty certain that silverspawn's simulation is correct (and also that randomizing the order is unnecessary and should yield the same result as looking at the first three non-T letters, though it looks like that has been confirmed by the simulation too). So let me see if I can get the actual reasoning correct:
The three T's have no effect on anything, so really we're just looking at four letters.
We know that a random two letters are P and a random one is K. But again, we can ignore these, because we only care about the one that we don't already know.
The probability of getting whatever letter we wanted (I forgot which one) given that it's not a T is 1/5.
So yeah, this is the same as my understanding of the two 5-sided dice problem. The way in which we get the information is not to re-roll until we get a 5 (Haddock's interpretation of the problem), but instead to look at what we already rolled, look at a random one of the dice (this is the same as always looking at the first die), and see that it happened to be a 5. The same thing happened in the game. We found out first that there are exactly three T's. Of the remaining four letters, we looked at a random three of them, and saw their results; then asked what the fourth one showed.
For Haddock's argument to be correct, we would have needed to re-roll the set-up over and over until we got a situation where we had 2+ P's and 1+ K, and then ask what the other one showed. This is subtly different, because it's not taking into account that we stumbled upon that information randomly. This would have been the correct solution if the situation was something like "any time 2 (or more) P's and 1 (or more) K is rolled, the corresponding roles will claim", and we wanted to find the probability of the non-claimer representing a P from there.
