Probability that colored balls are separated

Question

Say we throw $b$ blue balls and $r$ red balls uniformly into $n$ boxes. The probability that no box contains a red as well as a blue ball is then, by the inclusion exclusion principle:

$$p = \sum_{k,\ m} \binom{n}{k} \binom{n-k}{m} (-1)^{n-m-k} \left(\frac{m}{n}\right)^r \left(\frac{k}{n}\right)^b$$

I'm interested in upper bounding $p$. If we assume the red balls are thrown first, and without replacement, we can lower bound it as

$$\left(1-\frac{r}{n}\right)^b$$

Or 1 if both $r$ and $b$ are as big as $n$. Note $e^{-rb/n}$ is another good approximation, though neither a lower nor an upper bound.

Can you see any way to get a simple upper bound? I've tried to look into asymptotic expansions, but I don't see how I can be done.

Answer 1

Not what you are looking after, but a simple asymtoptic result would be: assume the number of blue (red) balls in each box are iid Poisson with $\lambda_B = b/n$ ($\lambda_R=r/n$), hence the probability that no box has red and blue balls would be

$$P \approx \left(1-(1-e^{-\lambda_B})(1-e^{-\lambda_R}) \right)^n=\left(e^{-b/n} + e^{-r/n} - e^{-(r+b)/n} \right)^n \tag{1}$$

I would bet that this is only useful for $n$ much bigger than $r,b$, and that it overestimates $P$, but it does not seem easy to prove that it's an upper bound.

Another approximation: the expected number of boxes with red balls is $E_r=n \left(1-(1-1/n)^r\right)$. Equating this (expected) number with the actual number, we get

$$P \approx \frac{(n-E_r)!(n-E_b)!}{n! \, (n-E_r-E_b)!} \tag{2}$$

A few results (ps comes from simulation, pa1 and pa2 are the above approximations)

  n  b    r   ps     pa1     pa2
1000 30  10  0.741  0.745  0.736
 500 30  30  0.165  0.183  0.168  
 500 80   8  0.278  0.309  0.275 
 500 20   6  0.786  0.791  0.782
  10  5   3  0.215  0.341  0.167

Answer 2

In case anyone is interested, this chart plots the seperation probability for $n\in[1,20]$, $r=2,b=3$. The dots are the correct value, the green line is $e^{-rb/n}$, the orange line is the lower bound from my original post, and the blue line is the upper bound from my comment to leonbloy.