The chance of failure would be 8 times that much because there are 8 pieces you could miss, which comes to about 12.7 per million.
But a uniform distribution is random, isn't it? This sort of phrasing is rampant in this thread, and I'm confused by it. How can a uniform distribution be considered non-random, or "less" random than another distribution?
Read the article for more details and examples.
Well, you're still assuming the colloquial definition of "random," which implies "uniformly random." Of course, we can have a random process that is not uniformly random.
The results of a uniform distribution, applied randomly a large number of times are less uniform than intuition predicts.
A little randomness is good, it keeps the experience from being monotonous, but there is a balance between uncontrollable and unpredictable annoyance and quirky interesting outlier behavior. A lot of why WoW did so well was because it often walked the fine line, and it did it everywhere. In that game crafting had percent chances of giving skillups, you had to level weapon skills with a percent chance per hit of getting a point, you had critical % chance, hit % chance, dodge %, parry %, block %, you had talents that gave you % procs for special effects. Tremendous amounts of RNG that most people never even noticed but kept them playing for so long in many respects.
I was bit by another random quirk in World of Warcraft. There was a long-running achievement called "What a Strange Long Journey it has Been", which took at least a year in real-time to complete, and you had to actively play during each of the ten or so in-game festivals and holidays, and do some quests and tasks during each. If you missed a festival, you had to wait another year to do it again, so if you were aiming for the achievement, you really wanted to do it in one go.
During their version of Valentine's, each player got a bag of heart-shaped candy, and the task you had to complete was to pull out at least one each of the eight different heart candies. But you could only pull a piece of candy once every hour or so, each time you pulled a piece you had a 1/8 chance to get a certain piece, but the holiday was time-limited, so you had about two weeks to complete it.
Sounds easy and fair, right? 1/8 chance, get all eight pieces, two whole weeks, easy! Except that there was one piece that I just never got. The piece that said "I LOVE YOU!". And as the time went by, I got more and more frantic, logged in more often so as not to miss any opportunity to pull a piece of candy, but no luck.
So, I did a quick bit of math. You can pull one piece every hour for two weeks, but sleeping, not playing, missing days etc, meant that I effectively pulled ~100 pieces. The chance of missing a certain piece is 7/8, so missing a certain piece 100 times in a row is (7/8)^100, which is roughly a little bit more than one in a million.
With ten million players all doing the same thing, there's going to be a number of them that will hit that "one in a million" chance, which means that whoever designed that part of the achievement didn't do their homework, didn't do the math. Because intuition tells you that 1/8 chance is plenty! Hundreds of tries, of course everyone will get all pieces! Except proper math tells you otherwise, and I was one of the "lucky" outliers.
(Later, they apologized and retroactively removed that part of the achievement, so I got my purple nether-drake mount without having to wait a year extra.)