Question

A certain box contains 200 particles of an ideal gas. how many times more likely is it to find the particles evenly split between the left and right halves of the box, than to find 160 particles on one side and 40 on the other?

Answer 1

Each particle has two possibilities: either it is on the left hand side or it is on the right hand side.

The probability of having them equally split is

P(100,100) = 200C100 / 2^200

where 200C100 is the binomial coefficient.

For the uneven distribution we have

P= 2 * 200C160 / 2^200

where the factor 2 comes in because we could have 160/40 or 40/160 as a division.

So the ratio of probabilities is

200C100 /(2 * 200C160 )

= 2.2 * 10^16

Hope this helps!