Assuming each rare occurs with an equal probability
But what decides which of the 25 possible rare treasures is going to drop? I've gotten some rare treasures only once. I've gotten other rare treasures 2, 3, 4, 5 times. I haven't gotten the rest of the rare treasures at all.
4 rare treasures in 100 picked up treasures sounds right to me.
Scenario 1:
If you got 4 new rare treasures in every 100 picked up treasures, you would have to pick up a total of 625 treasures to have all of the 25 rare treasures. 4% chance for a rare treasure applies here.
Scenario 2:
Your first 24 picked up treasures could be 24 unique rare treasures. Then, you pick up the next 600 treasures, none of which are the last rare treasure you need. The next treasure you pick up (#625) is the 25th and final rare treasure you need. 4% chance for a rare treasure still applies.
Scenario 3:
You've picked up 625 treasures. 25 of those were rare, but you got 12 of the rare treasures twice and 1 of them once. You keep playing and have now picked up 1249 treasures. You got 11 rare treasures you haven't gotten in the previous 625, but you also got them twice. The unique rare in the first 625 was picked up a second time. Treasure #1250 is the last rare treasure (#25) you need. 4% applies.
Scenario 4:
You've picked up 625 treasures. You've gotten rare treasure #1, 25 times.
You've picked up 1250 treasures. You've gotten rare treasure #2, 25 times. You now have 2/25 of the rare treasures.
You've picked up 1875 treasures. You've gotten rare treasure #3, 25 times. You now have 3/25 of the rare treasures.
etc....
You've picked up 15000 treasures. You've gotten rare treasure #24, 25 times. You now have 24/25 of the rare treasures.
In the next 25 treasures (15025, you get rare treasure #25. For whatever reason, you continue playing until you've picked up 15625 treasures.
In total, you've gotten each of the 25 rare treasures, 25 times. Each rare treasure occurs equally, and there's a 4% chance of a rare treasure dropping.
---
Naughty Dog says playing on harder difficulties gives "better" (rare I'm assuming) treasures. Hunter has no difficulty setting and since human opponents could be impossible to beat, the treasure "settings" are probably the same as crushing arena, explaining why people experience Hunter giving "better" treasures.
This system is lottery based as there is no predetermined requirement for a particular treasure to drop. The
chance is there, but there's no guarantee.