Assumption #1: you must be at least skill level 450 to have a chance for it to proc
Assumption #2: drop rate is probably 1 or 2%, to take the pessimistic approach I will guess 1%
Assumption #3: it takes between 30 and 35 fragments to solve troll projects so we will guess an average of 33
Assumption #4: you will likely have a bunch of troll tablets which basically count as 12 fragments each to make things easier
To find the probability of getting it you take 1 - P(event never occurs) = % chance. So if looking at your chance in two tries it would be:
1 - P(.99 x .99) = 0.0199 (about 2%)
So to get up to a probability of about 50% (a coin flip) you would need to solve about 69 troll projects
1-P(.99^69) = .5002 (about 50%)
If you want to get up to 90% chance you would need to solve about 230 troll projects
1-P(.99^230) = .9009 (about 90%)
Results:
1% chance = 1 troll project solved = ~33 fragments
50% chance = 69 troll projects solved = ~2,277 fragments
75% chance = 138 troll project solved = ~4,554 fragments
90% chance = 230 troll projects solved = ~7,590 fragments
99% chance = 460 troll projects solved = ~15,180 fragments