# Fortnightly prize question for 16 December 2014

Sometime in May 1936, Alan Turing sighed with satisfaction as he finished his paper “On Computable Numbers”, which is now famous as the start of computer science (though the first electronic computer was not built until 1943). Take a deep breath. How likely is it that you inhale an atom which was also in Turing’s sigh?

This is an example of a “Fermi problem”, a mathematical problem where you have to start by making a good guess at the information you need for the problem and then deduce a good guess for the answer, rather than being given exact information and getting an exact answer.

The new film “The Imitation Game” is loosely (very loosely: see review) based on Alan Turing’s life. He killed himself in 1954 after being arrested for gay sex and punished by chemical castration. 55 years later, in September 2009, the prime minister officially apologised to Turing.

No-one claimed the prize this fortnight, although Soner Hasan got near working it out. Oh well: exams, Oxford interviews, King’s Factor, end of term…

Solution: see http://www.hk-phy.org/articles/caesar/caesar_e.html.

That just tells us that on average there will be something like one atom from Turing’s sigh in each breath we take. In other words, it is fairly likely that you will inhale an atom from Turing’s sigh.

Year 13 Further Maths students will be able to go further and say that the number of atoms from Turing’s sigh in each breath follows a Poisson distribution Po(k), with k equal to about 1. If k=1 then the probability of inhaling at least one atom from Turing’s sigh is about 0.6.

If you breath in four times, the number of atoms from Turing’s sigh you inhale is Po(4), and the probability of inhaling at least one is about 0.98.

Even if your rough calculation is so far out that the average is half an atom from Turing’s sigh per breath, the probability of inhaling at least one of them in four breaths is about 0.86.