If you think that the dices show too many doubles and you have a bit of interest in statistics, I recommend to check out this post.
One important note: The number of doubles in a Backgammon game can NOT be modeled accurately with a binomial distibution. For a binomial distribution, you assume a fixed number of trials (dice rolls). But in a Backgammon game, the number of dice rolls will actually depend on the number of doubles. Higher proportion of doubles will be associated with shorter games.
In games where lot of doubles happened, you skipped more pips per turn, hence games will tend to be shorter and therefore a higher proportion of doubles will not be as rare as derived from a binomial model. Of course the relation between game length and number of doubles is more complex in reality, it even depends on play style. It is really hard if not impossible to find a proper probabilistic model for this.
The variance for proportion of doubles in a Backgamomon game should definitely be higher than indicated from calculations with the binomial distribution.
To make things much more intuitive, let's play a much simpler game: A solo game called "coinflip once or twice":
1st round: You flip a coin, if it shows heads, you win and stop the game. If not, move to 2nd round.
2nd round: You flip the coin again, if it shows heads, you win. If not, you loose the game.
Clearly the game length here depends on the actual results of the coinflips. (In Backgammon, when you roll a double (the higher ones: 3-3, 4-4, 5-5, 6-6), this also simultaneously will usually help you to win the game and usually shortens the game length.)
You can observe a few things when you play the coinflip game 100 times (I use "expect" here in the sense of a probabilistic expected value):
1. You will expect to win 75 of 100 games. You expect to win 50 games in the 1st round and 25 games in the 2nd round.
2. You expect 150 coinflips to happen over the course of 100 games. You expect 75 times to flip heads and 75 times to flip tails. So even though the rules of the game kind of bias towards heads, you still get equal probability of 50% for both outcomes. After all, you perform independent coinflips and each coinflip has 50% probability for either heads or tails.
3. If you calculate the proportion of heads for each of the 100 games, then the expected average of these proportions is 62.5%, not 50%. (You expect 50 of 100 games with only one flip that shows heads, these have 100% heads. 25 of 100 games show 50% heads, and the remaining 25 of 100 games show 0% heads. Taking the average (50*100% + 25*50% + 25*0%) / 100 yields 62.5%)
4. Among all the games that finish after one flip, 100% of flips are heads.
5. Among all the games that finish after two flips, only 25% of flips show heads and 75% of flips show tails.
The 2nd and 3rd point are very interesting. Heads will be shown in 50% of all flips, however, if you look at the average of proportions of heads in all single games, it is much higher than 50%. In the same vein, even though the proportion of doubles in all Backgammon dice rolls will be close to 1/6, that doesn't mean that the average of all final proportions of doubles from single games needs to be 1/6.
The 4th and 5th point show that you cannot simply apply binomial distribution to calculate probabilities for frequencies of heads and tails. You need to consider the characteristics of the game. In the example, short games have higher proportion of heads (100%) while long games have smaller proportion of heads (25%). In Backgammon, short games will tend to have greater proportion of doubles while long games will tend to have smaller proportion of doubles.
Conclusion: All in all, things are more complicated for Backgammon. But the example goes to show that not even the expected value of the proportion of doubles from the Backgammon single game statistics shown by BGA needs to be 1/6. The more interesting parameter would be the variance of these proportions, it should be higher than derived from a simple binomial distribution, making rare events of high proportion of doubles more likely than assumed at first glance.
What can I do if I still suspect the RNG to show too many doubles?
This is easy to investigate in general but tedious/limited on BGA. Pick a large number of played games. For each game, count how many non-double rolls happened until the first double happened. This number should follow a negative binomial distribution. https://en.wikipedia.org/wiki/Negative_ ... stribution In particular, the average number of non-doubles until the first double should be 5. We can also count the number of non-double rolls until 2 doubles happened. The average of these should be 10.
Edit: I included 4. and 5. and one paragraph about them to make my main point more clear.
One important note: The number of doubles in a Backgammon game can NOT be modeled accurately with a binomial distibution. For a binomial distribution, you assume a fixed number of trials (dice rolls). But in a Backgammon game, the number of dice rolls will actually depend on the number of doubles. Higher proportion of doubles will be associated with shorter games.
In games where lot of doubles happened, you skipped more pips per turn, hence games will tend to be shorter and therefore a higher proportion of doubles will not be as rare as derived from a binomial model. Of course the relation between game length and number of doubles is more complex in reality, it even depends on play style. It is really hard if not impossible to find a proper probabilistic model for this.
The variance for proportion of doubles in a Backgamomon game should definitely be higher than indicated from calculations with the binomial distribution.
To make things much more intuitive, let's play a much simpler game: A solo game called "coinflip once or twice":
1st round: You flip a coin, if it shows heads, you win and stop the game. If not, move to 2nd round.
2nd round: You flip the coin again, if it shows heads, you win. If not, you loose the game.
Clearly the game length here depends on the actual results of the coinflips. (In Backgammon, when you roll a double (the higher ones: 3-3, 4-4, 5-5, 6-6), this also simultaneously will usually help you to win the game and usually shortens the game length.)
You can observe a few things when you play the coinflip game 100 times (I use "expect" here in the sense of a probabilistic expected value):
1. You will expect to win 75 of 100 games. You expect to win 50 games in the 1st round and 25 games in the 2nd round.
2. You expect 150 coinflips to happen over the course of 100 games. You expect 75 times to flip heads and 75 times to flip tails. So even though the rules of the game kind of bias towards heads, you still get equal probability of 50% for both outcomes. After all, you perform independent coinflips and each coinflip has 50% probability for either heads or tails.
3. If you calculate the proportion of heads for each of the 100 games, then the expected average of these proportions is 62.5%, not 50%. (You expect 50 of 100 games with only one flip that shows heads, these have 100% heads. 25 of 100 games show 50% heads, and the remaining 25 of 100 games show 0% heads. Taking the average (50*100% + 25*50% + 25*0%) / 100 yields 62.5%)
4. Among all the games that finish after one flip, 100% of flips are heads.
5. Among all the games that finish after two flips, only 25% of flips show heads and 75% of flips show tails.
The 2nd and 3rd point are very interesting. Heads will be shown in 50% of all flips, however, if you look at the average of proportions of heads in all single games, it is much higher than 50%. In the same vein, even though the proportion of doubles in all Backgammon dice rolls will be close to 1/6, that doesn't mean that the average of all final proportions of doubles from single games needs to be 1/6.
The 4th and 5th point show that you cannot simply apply binomial distribution to calculate probabilities for frequencies of heads and tails. You need to consider the characteristics of the game. In the example, short games have higher proportion of heads (100%) while long games have smaller proportion of heads (25%). In Backgammon, short games will tend to have greater proportion of doubles while long games will tend to have smaller proportion of doubles.
Conclusion: All in all, things are more complicated for Backgammon. But the example goes to show that not even the expected value of the proportion of doubles from the Backgammon single game statistics shown by BGA needs to be 1/6. The more interesting parameter would be the variance of these proportions, it should be higher than derived from a simple binomial distribution, making rare events of high proportion of doubles more likely than assumed at first glance.
What can I do if I still suspect the RNG to show too many doubles?
This is easy to investigate in general but tedious/limited on BGA. Pick a large number of played games. For each game, count how many non-double rolls happened until the first double happened. This number should follow a negative binomial distribution. https://en.wikipedia.org/wiki/Negative_ ... stribution In particular, the average number of non-doubles until the first double should be 5. We can also count the number of non-double rolls until 2 doubles happened. The average of these should be 10.
Edit: I included 4. and 5. and one paragraph about them to make my main point more clear.
