Hey there! I understand your concern about the rolls, and we're here to shed some light on how the dice work in Backgammon to get a better since of what to expect.
First, let's look at the pure probability:
With a pair of six-sided dice, there's exactly a 1 in 6 chance of rolling doubles. That's the clear-cut mathematical probability.
Now, let's see what that means in a single game, where each player might roll the dice about 25 times:
In a game with 25 rolls, the statistically pure probability would predict exactly 1 in 6 rolls since there is six sides to a die. So that works out to:
4.17 doubles.
But dice are random, and in the small sample size of one game, you'll likely get doubles anywhere between
3 to 5 doubles one each side in 95% of games. You can certainly get some out side that range, after all this is randomness we're dealing with.
That range might seem wide, but it's perfectly normal when dealing with a small sample like one game. Over many, many games, the number of doubles should average out closer to the pure 1 in 6 probability. But in one individual game, the results can vary quite a bit.
It's like tasting a spoonful from a big pot of soup. That one taste might not perfectly represent the entire pot, but it gives you a good idea of the flavor. Similarly, one game gives you a snapshot of how the dice behave, but it might not perfectly match the mathematical ideal.
We've attached to this article a sample of the dice rolls from the game that is enough to actually do testing for those of you that are inclined to analyze them yourselves.
Here are the results of the analysis based on the data provided:
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Frequency Distribution of Die Rolls
Player 0: {1: 3223, 2: 3290, 3: 3256, 4: 3337, 5: 3207, 6: 3335}
Player 1: {1: 3297, 2: 3245, 3: 3298, 4: 3228, 5: 3236, 6: 3236}
Number of Doubles Rolled
Player 0: 1573
Player 1: 1574
Average Value Rolled
Player 0: ≈3.51
Player 1: ≈3.49
Average Time Between Rolls (in seconds)
Player 0: ≈2.06
Player 1: ≈2.07
Based on the analyzed data, here are some points that might be of interest:
Frequency Distribution: The distribution of die rolls appears balanced for both players. This suggests that the die rolls are fair and random.
Number of Doubles: Both players have almost the same number of doubles (Player 0 has 1573, and Player 1 has 1574), which is another indicator of fairness.
Average Roll Value: The average roll values are close for both players (around 3.5), further supporting the notion of balanced gameplay.
Time Between Rolls: The average time between rolls is almost the same for both players, around 2 seconds. This suggests that both players are taking roughly the same amount of time to make their moves.
I hope this helps you understand why the rolls might seem off sometimes. It's just the nature of probability and randomness, not something unfair or rigged. Happy rolling! 🎲