/sci/ - Science & Math

I'm fairly sure I've heard the explanation for this before, but I can't remember, so I'm hoping somebody can help me out.

I'll confine my question to the example of a fair, six-sided die, but the answer should be general.

Every time I roll a die, each roll is independent. Every time I roll the dice there is a 1 in size chance of getting a 4, say. BUT, taken as a group, the odds of a 4 never occurring drops as the number of rolls in the group gets larger.

So it looks like a paradox - if I roll the dice 20 times and a 4 doesn't show up, it sure feels like a 4 is more likely on the 21st roll, but I know it's still just 1/6. Even if I roll 1 million times and a 4 never shows up, it's still 1/6 odds.

What's going on here?