An Interesting Proof

Just found this out somewhere in my lecture notes. Of course, I'm going to modify it!

Do you believe that you have just inhaled an air molecule which Jesus exhaled in his last breath saying "It is finished" after more than 2000 years ago?

Our intuition tells us the probability of this is really small. After all, the numbers of molecules exhaled by Jesus was a tiny fraction of all the molecules available in the atmosphere. However, get ready for a suprising answer!

Let's build a model to see clearly.

Assuming:
1. There are n air molecules available in the atmosphere.
2. Jesus exhaled p air molecules.
3. The air molecules were distribute uniformly over the years.
4. We have just inhaled q air molecules.

Let's consider a single (randomly chosen molecule) in the atmosphere.
The probability that this is in fact a molecule that exhaled by Jesus is p/n.

Conversely, the probability that this is not a molecule exhaled by Jesus is 1 - p/n.

So, if we inhale 2 molecules, the probability that none of them was exhaled by Jesus is (1 - p/n)^2.

And, if we inhale q molecules, again, the probability that none of them was exhaled by Jesus is (1 - p/n)^q.

And, the probability that we did inhale a molecule which exhaled in his dying breath is given as:
P = 1 - (1 - p/n)^q

Alright! We now have a model and are ready for calculations.
Let's estimate the values of our variables.
The atmosphere has a lot of air molecules, n (approx.) = 10^44
The capacity of a normal human lung, p & q (approx.) = 2.2 x 10^22

Let's substitute in the values above into the equation!
P = 1 - (1 - 2.2 x 10^22/10^44)^2.2 x 10^22 (approximately) = 0.99

Wow! The result is so amazing right?!

The chance that we have just inhaled a molecule which Jesus exhaled in his dying breath is more than 99%!