This thread took a different direction than I hoped for. I edited my first post to make my main point more clear.
My main point was:
The number of doubles in a Backgammon game can NOT be modeled accurately with a binomial distibution.
This is exactly against my main point of the first post.
To be clear: If you take a number of random dice rolls from all your lifetime Backgammon dice rolls and you want to know the probabilities of how many doubles occur in these rolls, then you can take the above spreadsheet. However if you ask for probabilities of double frequencies in a specific game with that number of dice rolls, it's a different story. As the dice rolls are now structurally grouped together, they are not simply a pre-fixed number of independent rolls anymore.
This gets most clear if you think of very short Backgammon games. Imagine a game that finished after just 20 rolls overall (10 for each player). This game inevitably must contain a large proportion of doubles, otherwise it's impossible to have finished after just 20 rolls. So if you ask about probabilities for double frequencies among all valid games that finish after 20 rolls, it is different from asking what is the probability of certain double frequencies in 20 random dice rolls.
To be more clear: It is still very valuable to use probabilities from binomial distribution as an approximation for the actual probabilities of double frequencies. However, we should all be aware that these probabilites are not precisely correct. This is the most true for the extremely small probabilities of extremely rare events where the binomial model will be far off the correct answer. In the end, the true probabilities will depend on playstyle and there is no general way to calculate them easily.
(To be very clear: All of my points are about properly finished Backgammon games. Don't let me even get started about aborted games...)
