## The Breath of Death (or smart air)

Posted by Chris on August 19, 2010 – 5:44 pm

What is the probability that, in any breath that you take, that at least one molecule of it was exhaled by Archimedes in his last dying breath?

Assume that a breath contains 10^22 molecules and that the atmosphere contains 10^44 molecules.

August 19th, 2010 at 6:48 pm

I guess we assume that all of Archimedes’s last breath molecules remain in the atmosphere, i.e. not absorbed in the sea or in organic matter, and that they are evenly dispersed. Then any given molecule now in the atmosphere has a 10^22 / 10^44 chance of having come from Archie, i.e. 1 in 10^22.

Then, as I see it, this becomes like the probability of two people in a group having the same birthday. So the first molecule that I breath in has a 1 – 10^-22 chance of NOT having come from the great man, the second a 1 – ((10^-22)-1), and so on. Multiply the 10^22 terms of this series to get the probability of not catching one of A’s molecules. For such large numbers this would be close to 1/e.

So, the probability of catching at least one of Archimedes’s molecules would be near enough (for me at any rate) to

1 – 1/e.

August 20th, 2010 at 3:24 am

Hi Wiz. The birthday problem is more complex than this one. But your final answer is right.

On rereading the source, I note that it suggested 2*10^22 molecules in a breath. Using that figure, the probability rises from 63.2% to 98.2%.

August 20th, 2010 at 5:53 am

So…. if the atmosphere has 10^44 molecules, and a breath is 2*10^22 molecules, and there are 6bn+ people on the planet (breathing presumably), how many…. unique(?) breaths per person is that then? Starting from the first breath…. and that we all breathe in unison…?

August 21st, 2010 at 12:58 am

i’m guessing these molecules are argon.

(the ones in the atmosphere that doesn’t react with anything else)

September 6th, 2010 at 4:06 am

i guess the final answer would be 10^21

September 6th, 2010 at 12:06 pm

Hi sumit, probabilities cannot exceed 1, yet alone get to 10^21

July 6th, 2011 at 4:13 pm

The question referred to molecules, not atoms. So we’re talking N2, O2, CO2, etc. These multi-atom molecules make up about 990,636 ppm of the atmosphere (99.1%). The rest is made up of monoatomic Ar, Ne, He, Kr, etc. Over time the larger molecules break up and reasemble through a variety of mechanisms. Assume that over thousands of years they are well mixed and reassembled randomly. You might assume that perhaps 10% of the atoms, particularly C and N are locked up in plants, the ocean, and soil and have been replaced by other similar atoms. Recalculate!