How to Calculate the Average Number of Boxes with Different Ball Distributions?

[Ionophore]

Hi all,

I’m trying to distribute M balls into N boxes. Each box has two compartments (thus 2N compartments total), each of which can hold one ball. There are more compartments total than balls. What I would like to do is calculate the number of boxes, on average:

1) With no balls (2 empty compartments)
2) With 1 ball (1 full compartment and one empty compartment)
3) With 2 balls (2 full compartments)

My first hunch was to say that, because I just want to scatter the balls randomly, the probability of a compartment being occupied is M/(2N), and therefore the probability of a compartment being empty is 1-M/(2N). Then the probability of encountering a full box should be the probability that the first compartment is full x the probability that the second compartment is full, and so forth, giving me:

P_full = (M/(2N))^2

P_1/2 = (M/(2N))(1-M/(2N))

P_empty = (1-M/(2N))^2

But these don’t add to one! What’s going on?
Thanks in advance.

 

[mathman]

You need to double P_1/2, since the are 2 possibilities: comp. 1 empty or comp. 2 empty.