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Same question as "Distribution of $N$ balls numbered $1$ to $N$ with replacement", but without replacement:

An urn contains $N$ balls numbered $1,2,3,...,N$.

I draw at random $n$ balls, one by one WITHOUT replacement.

Let $X$ the smallest number, the largest $Y$ and $S$ the sum of all the $n$ numbers

How to compute:

  • the probability $P(X=x,Y=y)$ that $X=x$ AND $Y=y$
  • the probability that $S=s$
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    @Did Sure you cannot find more productive ways of contributing to the site?2017-03-20

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Here is the distribution of $(X,Y)$. The event $[x\le X,Y\le y]$ corresponds to subsets of size $n$ drawn from the set $[x,y]$ of size $y-x+1$. There are $\displaystyle{y-x+1\choose n}$ such subsets hence $ P(x\le X,Y\le y)=c{y-x+1\choose n},\qquad \frac1c={N\choose n}. $ Decomposing the event $[X=x,Y=y]$ thanks to the events $[x\le X,Y\le y]$, $[x\le X,Y\le y-1]$, $[x+1\le X,Y\le y]$ and $[x+1\le X,Y\le y-1]$, one gets $ P(X=x,Y=y)=c{y-x+1\choose n}-2c{y-x\choose n}+c{y-x-1\choose n}, $ which can be simplified to $ P(X=x,Y=y)=c{y-x-1\choose n-2}. $

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    Thank you Didier. What about the second question (probability that S=s) ?2011-05-12