What’s the chance you have drunk the same water molecule twice?

Short answer: For any given water molecule, the odds are basically negligible. But the odds that you’ve drank at least one water molecule twice are pretty much 100%.

Long answer: Think in terms of the numbers of water molecules on earth. In a cup of water there are about 1024 water molecules (100 g / 18 amu ~ 1024).

The total mass of water on earth is approximately 1024 g of water, which works out to about 1046 water molecules on earth.

So if you pick 1024 molecules out of 1046, put them back into the 1046 and mix them back up, and randomly choose another 1024, what are the odds you’ll pick at least one atom twice? We can approximate it in the same way we do the birthday problem: P = 1-e-n2 /2m where n=1024 and m=1046. Turns out this number is basically equal to 1, so the odds are almost certain that any two glasses of water will have at least one atom in common. This generalizes between every cup of water – in that cup of coffee you’re sipping right now, the odds are good that it has shared atoms with basically every person to ever live.

A Reddit AskScience thread

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