A box is filled out by $1,000$ balls. The box can be thought of as containing $V$ sites and $V$ balls, with $V=1,000$. The box is repeatedly shaken, so that each ball has enough time to visit all $1,000$ sites. The ball are identical, except for being uniquely numbered from $1$ to $1,000$.
What is the probability that all of the balls labeled from $1$ to $100$ lie in the left hand side of the box?
What is the probability that exactly $P$ of the balls labeled $1$ to $100$ lie in the left hand side of the box?
Using Stirling's approximation, show that this probability is approximately Gaussian. Calculate the mean of $P$. calculate the root mean square fluctuations of $P$ about the mean. Is the Gaussian approximation good?
Any insight is greatly appreciated.