Assuming you live in eastern N America or western Europe and are over 30 years old?
What’s worse: peeing a molecule of Kaiser Wilhelm II.
This will be very complicated. First you need the volume of that berg. Then calculate the volume of the water that the ice will melt and become. Then you have to estimate the volume of ocean water that this water mixes into, because it mixed only into the North Atlantic at first, but eventually some of it got to the South Pacific.
The really tricky part, I think, is figuring the evaporation rate of that ocean water, and how likely it is that those clouds would rain upon freshwater sources of drinking water, and/or rain upon any farms whose produce you might eventually eat. Plus, some other that evaporation will stay in the air - most of us don’t live in 0% humidity climates.
Finally, add up all the water that humans intake, either by drinking, eating, or breathing, and deduct that which is exhaled or stays in the body for cell growth, or exits via feces, mucus, and saliva, and what’s left is the urine.
Now, consider the total volume of all the urine you’ve peed in your life, and divide it by… beats the heck outta me. 
It’s going to be basically 100% unless the iceberg didn’t melt or evaporate.
Yeah, exact number calculations are hard, but chances are probably 100% you pee one out every time assuming they are more or less distributed around the globe by now.
Related, every glass you drink contains a few molecules of Caeser’s urine.
And
Avogadro says the probability is 100%, never mind your age or location. Every time you pee.
On the other hand, maybe not, as water molecules are constantly dissociating into OH- and H+ and recombining into different molecules, that is, water molecules, yes, but with different individual atoms. Perhaps no molecule of water from the Titanic iceberg remains with the same original hydrogen and oxygen atoms. Then the probability would be zero.
Somewhat related to the question of what the chances are that in every breath you take, there is a molecule that Caesar exhaled in his dying breath. I actually went through the calculations once and while I don’t remember the precise answer I came up with, it was basically that it was very likely there would be at least several molecules. Googling this just now, one calculation puts it at better than 98% that there’s at least one such molecule.
My guess is that a similar answer applies here, assuming a fairly uniform global distribution of the meltwater.
If homeopathy had any validity, I could sink an entire navy.
With pee?
I remember reading an interesting analogy for this a few years ago.
Basically, if you took a normal-sized glass of water out of the ocean and were somehow able to mark those H2O molecules for later identification - say, make them purple - and then poured them back into the ocean and took a really big spoon and stirred up all the oceans, rivers, lakes etc until they were all perfectly mixed, and THEN scooped another glass of water out of the ocean somewhere…
It would contain about 1500 of your original purple molecules.
You all forget that in photosynthesis plants break down water molecules and release the oxygen. It was wídly believed that the oxygen that plants release comes from the carbon dioxide but in a test where they marked the oxygene atoms with using isotopes it was found out that the oxygen comes from the water.
Then there’s water’s autocatalyst where 2 H2O converts to H3O+ and OH-. And that is happening everywhere there’s liquid water.
So basically all water molecules are broken down all the time so statistically there are no intact water molecules even left from the iceberg that Titanic collided, so the probabilty of you peeing one in is 0%.
The disassociation constant for water is 10 power -14
(sorry can’t thunk out how to do the scientific notation)
So while the number of free H+ in a litre of water (55.5M) is a large number, it is essentially insignificant compared to the number of H20 molecules
I was going to pee a molecule of Hitler until it found out I was circumcised and refused to come out.
When I was a wee tyke my father told me that ever glass of water I drank contained Jesus’ urine. I think I stopped drinking water for a while.
But they are exchanging those H+ ions constantly so you cannot say that each H2O molecule is not affected. Some may be but most are and that is why in the end the probabilty of you peeing the same H-O-H combination that existed in the iceberg is 0%.
You have a curious use of the term “most”.
So there is 1 H+ ion for every 550 million H2O molecules, and ignoring the inconvenience that you are fighting Le Chatelier’s principle and an equilibrium of just 10^-14, how many of the 3.34x10^29 water molecules in a litre of pure disassociating into H+/OH- would you consider to be fairly described as “most”?
The H+ concentration of pure water is actually Log10-7 M, which by a serendipitous scientific co-incidence means that the pH of water is 7.
But the rate at which the H-O-H molecules dissociate and recombine into H+/OH- is enormous: Very few of them at a given time are dissociated, but in a finite amount of time all have been recombined. As Wikipedia states on the article about self-ionization of water, (Mechanism):
Random fluctuations in molecular motions occasionally (about once every 10 hours per water molecule) produce an electric field strong enough to break an oxygen–hydrogen bond, resulting in a hydroxide (OH−) and hydronium ion (H3O+); the hydrogen nucleus of the hydronium ion travels along water molecules by the Grotthuss mechanism and a change in the hydrogen bond network in the solvent isolates the two ions, which are stabilized by solvation. Within 1 picosecond, however, a second reorganization of the hydrogen bond network allows rapid proton transfer down the electric potential difference and subsequent recombination of the ions. This timescale is consistent with the time it takes for hydrogen bonds to reorientate themselves in water.
The inverse recombination reaction
H3O+ + OH− → 2 H2O
is among the fastest chemical reactions known, with a reaction rate constant of 1.3×1011 M−1 s−1 at room temperature.
(Bolding mine)
So H20 disassociates fairly slowly, and re-associates very quickly.
I’m just trying to get my mind around the thermodynamics.
Every 10 hours every H20 molecule develops a dipole sufficient to self-ionize but the ionisation of water is endothermic?
If (random googling) the mass of the iceberg was 75 MMT and we have 1.4E24 g of ocean, then (4.2E12 mol / 7.8E22 mol) * 0.055 mol/ml
= 1.8E12 molecules per ml
Assuming no chemistry and even distribution, and ignoring water not in the ocean. None of which are safe assumptions.
Isn’t equal distribution a BIG assumption? I have a problem accepting that as a given. Is there some research to back this up?