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Imagine the following scenario: You have an urn with $r$ red balls and $b$ blue balls. The red balls have two hooks, the blue balls have one hook. Consequently, if $r=1$ and $b=1$, you would be twice as likely to fish a red ball rather than a blue ball.

What is the expected fraction of blue balls that you fish without replacement ($f_{bf}$) as function of the sample size ($n=r+b$) and the fraction of blue balls in the urn ($f_{bu} = b/n$)?

If I were to draw all balls, then $f_{bf}=f_{bu}$. If I were to draw one ball only, then $f_{bf}=\frac{f_{bu}}{2-f_{bu}}$. What if I draw somewhere in between 1 and $n$?

I can simulate the expected fraction (see below), but I am looking for how to calculate the formula. How would I go about that?

enter image description here

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  • $\begingroup$ @DouglasZare: We don't know exactly, unfortunately. Most likely there are in the millions. Other than the issue that the distribution is actually discrete, would it matter that much, though, or could we normalize numbers by the total (since we can get a decent idea about the ratios)? $\endgroup$ – Jonas Nov 4 '12 at 12:15
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Although you do not mention the words, I think you are looking for the expected fraction of blue balls drawn without replacement.

In general there is not a closed form, though there are approximations. You should read about Wallenius' noncentral hypergeometric distribution

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  • $\begingroup$ Thank you. I've added the missing words. Also, thanks for pointing out the distribution. Since I don't really know N in my problem, it looks like this is going to be tough. $\endgroup$ – Jonas Nov 3 '12 at 22:42
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    $\begingroup$ You can try the BiasedUrn package in R for help with calculations $\endgroup$ – Henry Nov 3 '12 at 23:29
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This past question is related.

If $r$ and $b$ are not too large, you can compute the distribution recursively in $O(rb)$ steps.

Let $p(x,y)$ be the probability that at some point there are $x$ red balls and $y$ blue balls left. If $x \lt r, y \lt b$ then

$$p(x,y) = \frac{2x+2}{2x+y+2}p(x+1,y) + \frac{y+1}{2x+y+1}p(x,y+1)$$

Replace the term with $0$ if $x=r$ xor $y=b$, and $p(r,b)=1$.

To compute the average for $1000$ draws from $1000$ red and $1000$ blue balls using this formula, $618.05$ red, took about $0.05$ seconds. Here is some C# code:

        double[,] p = new double[r+1, b+1];

        for (Int32 x = r; x>=0; x--)
            for (Int32 y = b; y >= 0; y--)
            {
                if (x == r && y == b)
                {
                    p[x, y] = 1.0;
                }
                else
                {
                    p[x,y] += (x==r)? 0 : (2*(double)x+2.0)/(2*x+2.0+y) * p[x+1,y];
                    p[x,y] += (y==r)? 0 : ((double)y+1.0)/(2*x+y+1.0) * p[x,y+1];
                }
            }
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