# N white and black balls and N boxes Probability

Given $N$ boxes , $N$ white balls and $N$ black balls chooses one box randomly and one ball from it randomly Find the probability of getting white ball.

• "$N$ boxes, $N$ white balls and $N$ black balls" is rather imprecise. Do you mean that there are $N$ white balls and $N$ black balls in each of $N$ boxes? That there are $N$ black balls and $N$ white balls, distributed among the boxes at random? So that each box has at least one ball? So that each box has exactly two balls? So that each box has one black ball and one white ball? These things will change the answer quite a bit, so until they're clarified, there's no real way for you to get an answer. – Cameron Buie Aug 4 '13 at 3:51
• Given $N$ new users posting $N$ new questions which are worded as demands rather than questions, what is the probability they are all homework problems? – Fixee Aug 4 '13 at 3:59

It depends on how the balls are distributed among the boxes. We will assume each box has at least one ball. Then we could put a white into $N-1$ of the boxes, and all $N$ black and the remaining white into the last box. The probability of white is then $\frac{N-1}{N}+\frac{1}{N(N+1)}$. This is close to $1$ if $N$ is large. Similarly, by an analogous distribution, we can make the probability of black close to $1$.
If on the other hand we put $2$ balls in each box by a process that does not pay attention to the colours of the balls, then by symmetry the answer is $1/2$.