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Suppose we have a $n$ balls in an urn labeled $1$ through $n$, and we draw balls without replacement. Suppose we draw a first ball , and then draw an additional $k$ balls uniformly at random without replacement. What is the expected number of balls which have a label larger than the label of the first ball?

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up vote 4 down vote accepted

Let $X$ denote the number on the first ball. Then $$ \mathbb{P}(X=x) = \frac{1}{n} $$ Conditioned on the number on the first ball, the remaining $n-1$ balls are split into two sets, those below $x$, there are $x-1$ of these, and those above, and there are $n-x$ of those. The number $Y|X$ of those above $x$ in the sample of size $k$ drawn without replaced follows hypergeometric distribution $\operatorname{Hyp}(k, n-x, n-1)$ with mean $$ \mathbb{E}(Y|X) = \frac{k(n-x)}{n-1} $$ Thus $$ \mathbb{E}(Y) = \mathbb{E}(\mathbb{E}(Y|X)) = \sum_{x=1}^n \frac{1}{n} \cdot \frac{k(n-x)}{n-1} = \frac{k}{n(n-1)} \sum_{x=0}^{n-1} x = \frac{k}{2} $$

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+1 for introducing me to hypergeometric distributions. – Austin Mohr Apr 26 '12 at 20:04
I knew there had to be a distribution for this. Bonus, the combinatorial definition of the distribution is even intuitive. – Joe Apr 26 '12 at 20:16

Hint: what is the probability that ball #$j$ has a label larger than that of ball #$1$?

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with replacement, the answer is easy; without replacement, it seems to depend on the labels of balls $2$ through $j-1$ – Joe Apr 26 '12 at 20:08
No, it doesn't. The other balls are irrelevant. All that matters is that you're drawing two different balls (#$1$ and #$j$) and that the two possible orderings of any particular pair are equally likely. – Robert Israel Apr 26 '12 at 20:30
@Joe: I suggest to make sure you understand this answer. It happens quite often that questions such as yours about expectation values can be answered using linearity of expectation without having to worry about any of the complications caused by correlations between different draws. Note how much simpler Robert's solution is than the others. – joriki Apr 26 '12 at 21:58
@joriki ...with an indicator variable for each ball $j$, and the expected number of balls with labels greater than that of ball $1$ is the sum of the expectations of the indicator variables... – Joe Apr 26 '12 at 22:45
@Joe: Is that a question directed at me? If so, I'm afraid I don't understand it. – joriki Apr 27 '12 at 0:12

If the first ball is number $i$, the urn is left with $i-1$ balls with smaller numbers and $n-i$ balls with larger numbers. The expected number in this case is clearly $k\frac{n-i}{n-1}$, provided that $k\le n-1$. The $n$ possible values of $i$ are equally likely, so the overall expected value is $$\frac1n\sum_{i=1}^n\frac{k(n-i)}{n-1}=\frac{k}{n(n-1)}\sum_{i=1}^n(n-i)=\frac{k}{n(n-1)}\sum_{i=0}^{n-1}i=\frac{k}{n(n-1)}\cdot\frac{n(n-1)}2=\frac{k}2\;.$$

You can see this even more easily by imagining that you draw the last $k$ balls first and then draw the ‘first’ ball: by symmetry it must on average be in the middle of the $k$ balls.

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The expected number is not (at least to me) clearly $k \frac{n-i}{n-1}$, although I do see that the probability of drawing the first good ball is $\frac{n-i}{n-1}$. It's not immediately obvious to me that drawing without replacement doesn't change the expectation. – Joe Apr 26 '12 at 20:21
@Joe: You’re picking a randomly chosen $k$ of the remaining $n-1$ balls, and you might as well imagine picking all of them at once, since the draw is without replacement. On average you expect them to reflect the $i-1:n-i$ split in the urn, so you expect $\frac{i-1}{n-1}$ of them to be below $i$ and $\frac{n-i}{n-1}$ of them to be above $i$. Since you’re drawing $k$ of them, you just muliply those fractions by $k$. – Brian M. Scott Apr 26 '12 at 20:36
When randomly selecting items from a population (either with or without replacement), the distribution of the $m$'th item selected is the same as that of the first. The conditional distribution of the $m$'th item, given the first, is the same as the conditional distribution of the second item, given the first. – Robert Israel Apr 26 '12 at 21:47

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