> And so there will be quite a few samples of a thousand that are higher than 66% 1-4. Yes, roughly half of them. But that's not the point. The point is that

> And so there will be quite a few samples of a thousand that are higher than 66% 1-4. Yes, roughly half of them. But that's not the point. The point is that the vast majority of those will be only slightly more than 66%. You won't see huge discrepancies. Taking your example, you would expect out of 1,000 samples for about 333 to be 5 or 6. Suppose you actually got 33 (or fewer). What would be the probability? It is 7 * 10^-125. That's 0.0000000[...]000007 with 124 zeroes before the seven. 330k samples is nothing compared to that improbability. You would expect to roll dice for longer than the lifetime of the universe to see such an unlikely event. Or you might conclude that the dice aren't fair.