The long run
Tap a button to roll a fair die. Start with 10 rolls, then add more. The graph shows what fraction of all your rolls so far were sixes. The dashed orange line is one in six. Watch the blue line: does it get closer?
What's going on
A fair die has six faces, so the chance of a six is 1 in 6. That is about 16.7%. But chance does not promise one six in every six rolls. In 10 rolls you might get none, or four.
What chance does promise is about the long run. The more times you roll, the closer the fraction of sixes gets to one in six. After 10 rolls it can be far off. After 10,000 rolls it is almost never more than one percentage point away.
Now look at the other number in the box: how many sixes you are off by. That number does not shrink. After 10,000 rolls you may well be 30 sixes away from a perfect one in six. But 30 out of 10,000 is tiny, and that is why the percentage still closes in. The die never "catches up". The early luck just gets swamped by all the rolls that come after.
Sheet: What "one in six" really promises. Related demo: two dice, many rolls.