Player's View: Probability of number of certain die on table given dice in hand

I'm trying to make some AI for a single player version of a dice game named Dudo. The relevant aspects are that there are six players with six dice each (which only they can see until the end reveal), and the goal of the game is to guess how many of a certain kind of die are on the table (so eight 3s, or nine 3s, or ten 3s etc.)

So as an example, if I have a 1 two in my hand, how do I calculate the probability there are six 2s on the table?

You know what your dice have, so you are asking the chance the rest of the players have the right number. In your example,the other players have $30$ dice and you want $5\ 2$'s. It is a binomial distribution. The probability in your example is ${30 \choose 5}(\frac 16)^5(\frac 56)^{30-5}$