D&D 5th Ed. - Let's untangle the Rogue math

[Yora]

This is pure statistics, so I want a separate thread for it, not clogging up the 5th Edition thread.

It's really only a very small rule that appears on the WotC website, so it's clear for discussion.

Rogues have the special ability that on all their specialized skills, the dice roll is always treated to be at least a 10. It's like being able to both roll and take 10 at the same time and pick the better number. This compensates for the fact that there are no skill ranks and everyone can have the same skill modifier as a rogue, regardless of a class or level.
What I want to know is, how much exactly does that improve the rogues chances?

For the sake of this calculation, let's assume we have a rogue with a Stealth modifier of +0 and an orc with a Perception modifier of +0, just so we can clearly see what we are doing. There are no other effects that affect the rolls. It really is 1d20 vs 1d20, the one with the higher number wins. In case of a tie, the status quo remains in place, which in this case means the Rogue is still not detected.

So how do we calculate the chances for the rogue staying hidden with just rolling 1d20 and compare that to rolling 1d20 with all numbers under 10 being treated as 10?

Spoiler

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Doodling with a geometric solution without doing any math yet, I astimate the chance to increase from slightly above 50% to slightly below 62,5%.