Question

2. Basic statistics) Consider an imaginary wall dividing this classroom into two rooms, A and B, of equal sizes. If there is only one oxygen molecule, the probability that you will not have the molecule is 1/2. Assume that the total number of the molecule is  . (a) Compute the probability that the room A is completely empty and does not have any oxygen molecule. Explain why you are still alive.

Answer 1

Suppose the total number of molecules in n and the molecules are identical because they are all oxygen molecules. There are two rooms. We need to find the number of ways these molecules can be distributed in Room A and B. The distribution is shown below:

Room A	Room B
n	0
n-1	1
n-2	2
n - m	m
n-n = 0	n

So, the number of ways is

0+1+2 +3+4+5..... +n

This sum is equal to n(n+1)

Out of these, there is only 1 way to put all the n molecules in B and 0 molecules in A.

So the probability that the room A is completely empty is

P = 1/(n(n+1))

In reality, the number of molecules is very large and the number can be approximated as infinity

Hence

So the probability that the room A is completely empty is 0 in reality which is why there will be still some molecules left for you to breathe.