Re: jars and marbles again...

>Hi,
>
>Consider the problem of putting M identical marbles >randomly in J jars. I
>was asked the expected number of non-empty jars. Using >indicator variables
>I was able to calculate this expected number and the >corresponding
>variance.
>
>Although this was good enough, I'd like to know if it's >possible to
>calculate the exact distribution of the number of non->empty jars.
>
>Thanks,
>
>Boi

Hi,

let X be the number of _empty_ jars.

Then,

P(X=i) = C(J,i)*C(M-1,M-(J-i))/C(J+M-1,M).

This can be explained as follows:

First, choose i jars out of J which are meant to remain
empty (C(J,i) possibilities).
Then, distribute one marble on each of the remaining J-i
jars. You are left with M-(J-i) marbles which can
be distributed on the J-i jars
(C(J-i+M-(J-i)-1,M-(J-i)) = C(M-1,M-(J-i)) possibilities).
Now, put the possibilities together to get
C(J,i)*C(M-1,M-(J-i))possibilities to distribute M
marbles on J jars such that exactly i jars are empty.
Finally, the total number of possibilities to distribute
M marbles on J jars is C(J+M-1,M).

The probability that exactly i jars are non-empty
can be easily deduced from this result.

Best wishes
Torsten.
.

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