n(n+1)/2 * 1/400
for a range where n = 11,+1 vs +8 = 16.5 percent
for a range of where n = 12, +1 vs +7 = 78 out of 400 (19.5 percent )
this is how often extraneous factors are assumed to interject.
Admitting my math might be bad this morning...
I think you might be or maybe I am misunderstanding your post. How can n = 11 if it is +1 vs. +8? Wouldn't that be a difference of +7? With a difference of +7, I got a "lose" percentage of 19.5% and ties only 3.25%.
Anyway, for those how feel something like I do, I want to focus on the issue of expertise (our stealthy rogue) and passive perception, so let's look at some actual numbers.
Did you know that the typical rogue with expertise in RAW (assuming some reasonable boosts to DEX over his career) needs only an average of 5.6 (4.6 ties) on his check to beat foes with a CR equal to his level (or in the case of CRs 21+, when he is level 20)? Let's round that up to an even 6, so there is only a 25% chance the rogue will be noticed. This is looking at over 2000 foes, by the way.
Now, you might think, "Well, that's fine, he has expertise, after all." True, but for any rogue who plans on being stealthy, he will most likely have expertise in it, meaning more likely than not he won't be spotted at all. And of course, against the majority of foes that won't be considered hard or deadly (CR equal to the rogue), the number he needs is lower.
Oh, and this assumes we grant every foe proficiency in perception
... which is incredibly generous since many creatures and NPCs don't have proficiency in perception. So, really, his needed average roll is even lower...
This is also assuming no buffs or advantages of any kind, which is hardly likely, especially against a "high perception" target! So, his number is even lower still...
Now, take away expertise, and that number increases to an average of 8.4 to beat passive perceptions against equal CR targets (with the same assumptions of every foe getting proficiency in perception AND no buffs, etc.). Rounding up to 9, there is a 40% chance to notice the rogue.
Considering the assumptions, that is more reasonable to me, especially when you consider his odds will be even better against most foes in most circumstances.
At our table, we see it time and again that our rogue bypasses nearly everything via his expertise in stealth. When I am DMing I don't want this. Expertise is TOO good under most circumstances. It is great RAW when it is really needed, but it seems like it offers too much benefit over all. The same can be seen for expertise and just about any skill. Take my own character, with expertise in Arcana (she is a rogue/wizard). When it comes to arcana checks, she makes just about everything, especially when given buffs or advantage due to the sorcerer (with arcana proficiency) in the group "helping".
Our chances would be pretty good under most conditions even without expertise, and expertise is taking too many things that should be a challenge to some degree and making them practically pointless.
I cry out again, "DOWN WITH EXPERTISE!" LOL