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No one but the devs know what the actual encounter rates are for each mon, but the extinct mons have a 1/20 chance (1/30 for tyrants) to be added to the pool of mons you can enouncter when you trigger a wild encounter, then if that check is successful, you have to compete with EVERY other mon in the pool for that zone for it to actually spawn.
So lets say Darine has a 5% encounter rate (which is probably 5-10x more than what it actually is, but let's be generous) then that would be (1/20) for a 5% chance times (1/20) for pool chance = 1/400 chance to actually encounter it.
But realistically, the encounter rate for extincts its probably around the 1% mark or lower, so 1/2000 would be a safer bet.
For switch, since they are a patch behind, it's 1/100 for both extinct and tyrant.
Assuming the 1% spawn rate + 1/30 pool encounter rate, then if we're really generous and assume you have 380/381 nexomon and this is your alst mon you need, your cosmic chance is 1/1855.
So thats: (1/100)*(1/30)*(1/1855) = 1/ 5,565,000 to find the last mon you need as a tyrant, in cosmic.
Odds only get worse the less mons you have seen and caught.
That is a little on the dumb side. I get sick of grinding hours on end for 1 monster to never spawn in games like this