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From: Fons on 16 Feb 2010 14:33
Given M marbles to be randomly put in N jars, I needed the probability
that at least one of the jars remains empty. With some inclusion/
exclusion, I seem to get a correct result :

P(M,N) = sum(binomial(N,i) * (1 - i/N)^M * (-1)^(i+1),i=1..N-1)

I was wondering if there is a more closed form of this result (and what
the corresponding, more straightforward, reasoning behind it would then
be).

Thanks,

Kobboi
From: Gerry Myerson on 16 Feb 2010 17:11
In article <4b7af326$0$2853$ba620e4c(a)news.skynet.be>,
Fons <fons(a)fons.invalid> wrote:

> Given M marbles to be randomly put in N jars, I needed the probability
> that at least one of the jars remains empty. With some inclusion/
> exclusion, I seem to get a correct result :
>
> P(M,N) = sum(binomial(N,i) * (1 - i/N)^M * (-1)^(i+1),i=1..N-1)
>
> I was wondering if there is a more closed form of this result (and what
> the corresponding, more straightforward, reasoning behind it would then
> be).

I think inclusion-exclusion is the standard way to do this problem
(or the complementary problem of finding the number of onto
mappings from an M-set to an N-set) which suggests that there
isn't a closed form (unless you accept Stirling numbers as a closed
form).

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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