Looking at a standard double-six dominoes set that has 28 tiles 0:0 through 6:6 how would we calculate (or estimate) the number of possible moves in a single game? For this so far I've come up with:
Before game tile selection (4 players): each player draws 7 of 28 tiles so 28 choose 7 = 1,184,040 21 choose 7 = 116,280 14 choose 7 = 3,432 7 choose 7 = 1
Added up = 1,303,753 possible combinations of tile draws before the game starts (assuming that adding the combinations is allowable statistically?)
The first play in the game is whoever has the 6:6 tile puts it down. The next play, goes to the next player and they play if they have a tile with a 6 on it (0:6 to 5:6). So that player will have some probability of having from 0 to 6 of those possible tiles and therefore 0 to 6 possible moves available to them.
How do we work out the probabilities of the player of having from 0 to all 6 of the remaining x:6 tiles?
We now know after player 1 played 6:6 that there are 6 X:6 tiles left out of 27 but what is the probability that player 2 has from 0 to all 6 of the remaining x:6 tiles?
Knowing that 1 of 28 tiles has been played there are 27 choose 7 ways he could have drawn his hand and 21 choose 6 ways of drawing a hand WITHOUT any of the x:6 tiles. So the chances of drawing none of the x:6 tiles would be 21 choose 6 / 27 choose 7 = ~6% ? And then continuing on for the subsequent number of possible x:6 tiles in hand?
The game is then played until one person plays all their tiles and their hand is empty.
If a player has no corresponding tiles they skip to the next player and keep skipping until someone could make a play.
I have no idea of the above calculations are correct or not, but looking for a some direction or help with how to go about figuring this out or a way of thinking about the game statistically. THANKS!