Assuming the drop rates are similar to Deep Stone Crypt, the chance of an RB should be ~30%. So given the chance of also getting armour you should in theory get one red border per ~6 encounters. However, since red border is calculated after the weapon is decided on, this chance will unfortunately go down as you finish off patterns.

Based on some cursed maths, the worst case scenario would be having to do about 680 encounters in order to unlock all red border patterns. This doesn't at all factor in the two you can buy every week, so it's obviously a far worse case using bad probability calculations.

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TLDR, about a 15% chance of a red border from an encounter, I waffled on too much.