Careful there, the sum should approximate a normal distribution (as should the average since a normal distribution divided by an integer is still a normal distribution). Also, the count of any of the possibler esults will approach a normal distribution (the number of 7’s rolled will be go towards a normal distribution around 1/6 the number of rolls as the number of rolls goes to infinity). The law of large numbers only works on sums.
The distribution of the rolls of a pair of dice is known and easy to calculate. You should expect 1/36 rolls to be snake eyes. If the results were normally distributed with a mean of 7 and a standard deviation of close to 2.04 just over 1/20 rolls would be snake eyes.
But you still can’t do this without reference to the manufacturing process. The distribution of number is unlikely on a fair die, but if the chance of the manufacturing process producing fair dice was 99.9999% then that would heavily outweigh the observed data - the data isn’t convincing enough to make me think I got the 1 in 1,000,000 die that isn’t fair. Of course we would never guess that hand forged dice have such a low defect rate. But if they were casino dice, I don’t think these rolls would give us much of a reason to think they were unfair.
