How can one explain the Kelvin Water drop experiment without assuming the container to already possess a positive charge? Kelvin Water Dropper experiment refers to the electrostatic generator invented by Lord Kelvin, which uses falling water to generate a voltage by electrostatic induction occurring between interconnected, oppositely charged systems. 
http://en.wikipedia.org/wiki/Kelvin_water_dropper
 A: The way I have heard it explained is not by the container but the water drops themselves. Statistically there is no way you can get a perfectly neutral water drop every single time. Eventually you will get a drop with a charge of 0.000000001 coulombs. This tiny imbalance is enough to set the experiment in motion into a positive feedback system.
You can think about it this way, imagine a factory that makes steel balls that have a mass of 100 grams each. Is it reasonable to think that every single ball from the factory will have a mass of exactly 100 grams and not vary to even 100.0000000000000000001 grams? Electrical charge functions the same way. No macro size object has a perfectly neutral charge. Even the earth ground in circuits has a charge, it is just that the actual charge is so insignificant that you can almost always consider the ground potential to be 0V.
A: There must be some asymmetry at the start.   It can go either way, positive or negative.  It's like balancing a razor blade on edge, and "falls" one way or the other.
But, under humid conditions, the Kelvin generator gives zero output.  Build it, turn on the water, and nothing happens.  This occurs because a many-megohms resistance appears across all the insulators.  Surface contamination is slightly conductive.   Then, even if there is small electrical noise in the system, any asymmetrical voltage will decay away through the resistance faster than it builds up.
However, there is always a threshold output-voltage below which the output decays to zero, and above which the output rises exponentially.
In somewhat-humid conditions, this threshold voltage is usually low enough that the normal charge on human bodies may give the metal parts an electrostatic inductive "kick" to exceed the threshold and start up the generator.   (Human bodies normally have a few hundred volts wrt ground, caused by walking on floors, touching against chair seats, etc.)
With higher humidity, sometimes we can "kick" the generator by charging an inflated balloon upon wool or hair, then waving the charged balloon near one side of the Kelvin generator.
PS
The ultimate insulator for DIY electrostatic device is thin nylon fishing line which has been cleaned of any fingerprint contamination.   With the metal parts suspended by fish line, many kilovolts can persist even over tens of hours, even with humidity high enough to produce droplets of condensation on the nylon line!   No other insulator has such a small surface area and such a long leakage path.
PPS, in Hunger Games II, see Tool Effects giant octagonal Kelvin device.  Built by G Finnin., I had some input on their design.
A: To include many other, clear & technical descriptions of this phenomenon, I should add that this 'generator' can be described as a bi-stable circuit, much like those used in most digital systems on earth.
It might also help to mention what could be called the 'Power Supply' path, which consists of the contiguous water($\mathbf CW$) (& electrostatic) path between the two drop formation($\mathbf DF$) areas (& any contiguous water source volume).  Any slight imbalance in the charge on the metal conductors is reflected in an imbalance in the static voltage in the CW path, caused by electrostatic/capacitance displacement of free protons$^+$ & electrons$^-$ (p$H^+$ and p$e^-$ of 7) from the DF areas, by any non-neutral charge on the so called 'induction rings'.  ('Induction' wrongly imply that the moving charge makes a relevant magnetic field here.)
(The well known "$\mathbf{p}H^+$" refers to the $\mathbf{p}$robability of free $\mathbf H$ydrogen ions (==protons$^+$) in neutral water (pH7=$\mathrm 10^{-7}$).  Naturally, for each free proton$^+$, there 'should' be a free electron$^-$, somewhere nearby.)  
In this environment, the static charges in the water, are pushed by the static ring charge. The charged drops are separated (by DF), pulled away by gravity into the buckets, & the charge trapped in the drops, spread to the metal conductors & rings.
