The 95% probability of being within 1 X RNP is a statistical probability. The 5% excursion may occur on the first RNP AR approach or it may not occur until the 100,000 approach. And this probability is for the entire fleet (world-wide) of RNP AR aircraft, not any given aircraft. So, it will occur to some aircraft in the fleet and not to others.
Further, this probability is not for GPS, or for that matter GNSS, because performance based navigation is considered to be sensor independent. As some others have said, the position used for the navigation solution goes through quite a process in an RNP AR certified aircraft.
This is not like lousy VOR radials are even more lousy NDB bearings. Odds are that you could fly the VNKT RNAV (RNP) Rwy 2 approach every day of your 35 year career and be on, or within a few feet of centerline every time.
And, to not be on, or within a few feet of center without having received an "Unable RNP" message is quite remote.
But, the engineers must deal with statistical theory when providing RNP containment areas without any secondary obstacle clearance areas that conventional PANS-OPS and TERPS have. Thus, the RNP AR containment area is 2 X RNP. And, with lower RNP values other mitigations must be included in the navigation solution, including EGPWS with enhanced database capabilities in an obstacle rich environment.
Finally, RNAP AR approaches are not "all weather" procedures, unlike fail-active CAT III autoland. RNP AR approaches have as their primary purpose the avoidance of terrain at terrain-rich locations. And, there are airports where it simply is not possible to design an effective RNP AR approach because the terrain is simply too onerous.