Question about molecular-level spontaneity?

Question

A container contains 2 mol of H2 and 1 mol of O2. Write down the probability that all the H2 is in the leftmost 2/3 of the container and all the O2 is in the rightmost 1/3.

I don't need a numerical answer for this, but I'm having trouble conceptualize a correct expression that will solve this problem.

Thank you!

Answer 1

For each molecule of H2, there is a 2/3 probability that it will be in the leftmost part. So for 7 independently moving molecules, just for example, the probability will be (2/3)^7, and for 2NA independently moving molecules, it will be (2/3)^2NA, where NA is Avogadro's number.

For your oxygen molecules, the probability is (1/3)^NA

For both these things to happen at once, the probability is got by multiplying those two separate probabilities together.

So, as you can see, it's just not going to happen.