Ten identical particles are moving randomly inside a closed box. What is the probability that at any given point of time all the ten particles will be lying in the same half of the box ?

The answer depends on how the sample space is counted. If the ten particles are treated as indistinguishable and only the occupancy numbers of the two halves matter (a Bose-like counting where swapping particles does not generate new microstates), the possible occupancies of one half are 0,1,2,...,10 — eleven equally possible outcomes — and the two favorable outcomes are 0 or 10, giving probability 2/11 [1]. By contrast, treating particles as distinguishable (independent placements) yields probability 2·(1/2)^10 = 1/512, but that uses a different microstate counting assumption. The question’s phrasing (“identical particles”) points to the indistinguishable counting and hence 2/11.

Sources

1. [1] https://courses.physics.illinois.edu/phys213/sp2013/lectures/lecture7.pdf