Aug 25, 2008

Large numbers are truly incredible

Here's a favorite example: every time you drink a glass of water, the odds are good that you will imbibe at least one molecule that passed through the bladder of Oliver Cromwell. It's just elementary probability theory.
--Richard Dawkins, The God Delusion, pg. 366

That's such a fascinating claim I had to do the math myself. Obviously, the water cycle is too complex to model exactly, but this is a guesstimate after all. To make things extra hard, let's bias all the numbers by rounding them to weaken the chances of this happning.

What number shall we use? How Stuff Works says there are 326 million trillion gallons of water on Earth. Let's round that up to 10^21. Let's assume an 8 oz glass of water for this experiment. And, according to the Wikipedia birth and death dates for Oliver Cromwell and a date subtractor, Cromwell lived 21,681 days. Again, let's make this more challenging, by assuming he only peed 1 fl oz a day. (One source I found said people pee 1.5 - 2 liters a day!) The rest are just conversion factors, which are not exact and of course depend on the temperature of the water. But the difference turns out to be many magnitudes, so the conversion factors only need to be correct to within one magnitude for the premise to hold.

1. How many mols H2O are in an 8 oz glass?

8 fl oz * 29 grams / fl oz * 1 mol H2O / 18 grams = 12.8 mol H2O, which we'll round down to 10 mol H2O

2. How many mols H2O are on the Earth?

10^21 gallons * 3.78 L / gal * 1000 grams / L * 1 mol H2O / 18 grams = 2.1 × 10^23 which we'll round up to 10^24 mol H2O

3. How many mols H2O did Oliver Cromwell pee?

21,681 days * 1 fl oz / day * 29 grams / fl oz * 1 mol H2O / 18 grams = 34,930.5 mols H2O which we'll round down to 10^4 mol H2O

So what does that give us? The percent of Cromwell's pee to the total water of the world is: 10^4 / 10^24 or 1 part per 10^20. That seems like a pretty small percentage. Yet in one glass of water, there will be

10 mol H2O * 6.0221415 × 10^23 molecules / mol = 6.02 * 10^24 molecules

So from a sample where we have a 1 in 10^20 chance of getting a molecule of Cromwell-water, we take 6 * 10^24 samples. Heck, by these probabilities, we'd have a likelihood of getting roughly 6000 molecules of water that once passed through Cromwell's bladder. I personally doubt that all the world's water mixes evenly, and most of it stays at the bottoms of the oceans, making the pool of drinkable molecules drastically smaller. And I believe the Cromwell probably peed more than 1 fl oz a day. And although I certainly concede the possibility that many of the water molecules that Cromwell passed may no longer exist because they were broken up by photosynthesis, I think it's still possible to say that "the odds are good".

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