Poissonization. When $N$ balls are placed in $M$ urns we get a multinomial distribution. For many calculations it's easier to consider the alternative (and apparently more complex) model in which the number of balls is random: $N$ is not fixed, only its mean. The models are (asymptotically, in some some sense) equivalent.

In the same vein, when computing asympotical statistics for head/tail runs of a sequence of $N$ random coins, it's sometimes convenient to consider that $N$ is random. For example.

In a similar vein (I guess the analogy has been noted somewhere), in Statistical Physics, the Canonical ensemble seems more complex than necessary (the kinetic energy is not fixed, as it is in the seemingly simpler and more natural microcanonical ensemble); actually, the Canonical ensemble is much easier to deal with. A similiar consideration applies when going from the Canonical to the Grand-Canonical ensemble, in which also the number of particles is not fixed.