Consulting the Codex…
RS3 · Drop Rate Calculator
Enter any drop rate and a number of kills to see your real chance of getting the drop — plus how many kills it takes to reach 50%, 90%, and 99%.
Each attempt is independent, so past bad luck never changes the next roll — the drop rate stays 1/128every kill. “50% chance by” is the median: half of players will have it by then, half won't.
The chance of getting at least one drop in N kills at a rate of 1/X is 1 − (1 − 1/X)^N. For example, at 1/128 over 128 kills the chance is about 63.4%, not 100% — each kill is an independent roll.
No. Each kill is independent, so the drop rate is exactly 1/X on every attempt regardless of how many times you have missed before. RS3 uses fixed-rate drops with no pity or bad-luck protection unless a specific boss is documented to have a threshold.
Roughly 0.693 × X kills, where X is the drop-rate denominator. At 1/128 that is about 89 kills for a 50% (median) chance — half of players will have the drop by then and half will not.
Expected drops is the average number of that item you would get over your kills: kills ÷ rate. Over 256 kills at 1/128 you would expect 2 on average, though any individual run can vary a lot.
Drop-mechanics references courtesy of the RuneScape Wiki (CC BY-NC-SA 3.0).