It's straightforward(!) to figure out the odds of not hitting a hand
like 4 of a kind (4oak) after N number of tries. The odds of hitting
4oak on your next hand is about one in 2361. That means that the
probability you will miss after one hand is 2360/2361. One chance to
hit, 2360 chances to miss.
The probability missing after two hands is 2360/2361 x 2360/2361.
Still not very likely to make 4oak, but slightly better. Two long
shots are better than one long shot. But when you play a large
number of hands, say 2361, you have a much better chance, but NEVER
guaranteed to hit.
So what are the odds of not hitting after 2361 chances (one cycle)?
The probability you will miss after 2361 hands is 2360/2361 to the
2361 power. According to ecalc.com, that answer is .36780. Call it
one chance is three that you won't get any 4oak by then. Two chances
in three that you will hit at least one. Maybe more than one.
Let's consider an unlucky run of 10,000 hands with no 4oak. What's
the odds of that? 2360/2361 to the 10,000 power is .01446. Call it
one chance in 98. Not likely, but it can happen.