This arose out of an online game, and both exact answers and approximations would be greatly appreciated.
$280$ balls are randomly put into $56$ bins such that all bins contain exactly $5$ balls. Among all $280$ balls, $2$ of them are colored red, $2$ of them green, $2$ of them blue (the rest can be considered uncolored, or any other color). I want to know the probability that, among the $56$ bins of balls, there exists $\ge 1$ bin which satisfies the following condition: the bin contains balls of at least two colors among the colors red, green, and blue.
For example, if we use "O" to denote a ball that is not colored as red, green, or blue. $\{R, B, O, O, O\}$ and $\{R,R,G,O,O\}$ are bins that satisfy the condition, while $\{R,R,O,O,O\}$ and $\{B,O,O,O,O\}$ do not.
My Attempt
I think an exact answer can be arrived (using multinomials), but I could not proceed beyond writing out the denominator. I decided to do a Poisson approximation, where $n = 56$ and $p$ is the probability that, when we randomly sample $5$ balls out of $280$, the $5$ balls satisfy the condition.
I calculated $p$ as follows:
$$ 1 - \frac { \binom{274}{5} + \binom{6}{1} \binom{274}{4} + 3 \binom{2}{2} \binom{274}{3} } {\binom{280}{5}} $$
And from then on I used $\lambda = n p$ and used $1 - e^{-\lambda}$ as the final probability that there exists $\ge 1$ bin which satisfies the condition.
- Is my $p$ correct, or did I count it wrong?
- Can I use Poisson approximation here? I know that Poisson can be used for weakly dependent events, but I am not sure this qualifies.
Edit: The Randomization Process
I realized that I probably should have stated the randomization process. The original game was randomized by randomly sampling 5 out of 280 into the first bin, 5 out of the remaining 275 into the second, 5 out of the remaining 270 into the third, and so on.