"Outstanding Potential", as in something the IV Judge says, is when the sum of a Pokemon's IVs is between 151 and 186 (31 in all stats).
All legendary Pokemon have three 31 IVs, right off the bat. This means that for those three stats, they add up to 93. This leaves 58 points left over to have at least 151 IV points total.
The probability of having your remaining 3 IVs add up to 58 or higher is 8376/29791 on one Pokemon, or 587638181376/26439622160671 for all three.
Percent: 2.222566486786309577404580666427% (2.22%)
How Sempiternus did it
Scraf asked me to calculate it after he couldn't, so here's just a quick overview of what I did to get that answer.
31^3 is the total amount of combinations you can get using 3 IVs.
I then proceeded to find the amount of combinations to get an IV sum of 58 - 93 (inclusive). Calculated the combinations for 58 (636) and so on - don't worry I didn't calculate individually for every single one. Derived a pattern that occurred and used that to solve the rest of it, then added everything up.
The Pattern relates to Triangular Numbers + some other numbers that were present in the calculations. I won't go into detail, since it was messy, and I'm too lazy to scan over my paper that I used.
Anyway, so yeah, I arrived at 8376 possible combinations for 58 - 93, assuming I didn't make any addition errors. That sum goes over 29791, which is 31^3.
Furthermore, to calculate the chance for 3 Pokemon to be under those conditions, you Cube that sum as well, thus giving you the answers above.
However, credit to Scraf for attempting and he did make some calculations with me during the process.