I am not sure though, but the significance appears as-
The stated scheme in the long run is equivalent to distribute M distinguishable particles
in two urns where each particle has probability 1/2 to go into an urn. Pai(i) is the probability that one specified urn will contain i particles. That is, in the stated scheme, whatever be the initial distribution of particles in the urns, in a long run they will be distributed as in case of a Binomial distribution.