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If you kill all the Champions, you get a Platinum score. You don't have to kill them, but it's generally advisable.
But as for the drops, I think the chance of an exotic dropping is about 20% (although I could be wrong about that). Regardless, 3 runs with no exotic drop is perfectly normal.
You also want to watch what the Elemental Burn is, and what the shields are going to be.
If there's two types of shields, I'll try to account for that when I'm picking Energy and Power weapons.
And if you're farming for gear, it's probably better to just run regular Legendary Lost sectors rather than Master, the drop rate isn't that much lower and they're easier to run quickly.
In Master LS, you have a max 33% chance of an exotic, and Legend is 25%. I'd recommend doing Legend instead since the faster completion rate offsets the lower odds.
So yeah, you won't get an exotic every run.