One thing I'm not seeing is that we tend to look at probabilities being 'fair' - If I roll a die 6 times, and it comes up 1, 2, 3, 4, 5 and 6, that feels 'fair' because each face has an equal probability to roll.

However, if I roll the die 12 times and it comes up with three 1s, three 2s, two 3s, two 4s, two 5s, and no 6s, that feels unfair - and it feels like we should roll two 6s to make it fair and match the probability.

However, probability doesn't really behave that nicely in the small scale. If I roll that die thousands and thousands of times, yes the results will be roughly equal per die face, but that doesn't mean that each set of 6 rolls maintains the large scale probability.