Someone may in principle know the exact velocities etc. of the balls or whatever is used to pick the random numbers in advance. But if we do not know them, we may describe them by "probabilistic distributions" which means that we assign every possible velocity and position of all the balls, and therefore every possible outcome of the lottery, some probability (or probability density).
However, probabilities and probability densities may be used even in classical (non-quantum) physics, to quantify our uncertainty and partial knowledge about the state of the physical system (the balls). Quantum mechanics is more "novel" than just probability distributions because the probabilities in quantum mechanics are calculated from some complex "probability amplitudes" by squaring them and the probability amplitudes are matrix elements of linear operators on the Hilbert space, and so on, so your picture seems to be missing all this actual "beef" of quantum mechanics.
For all practical purposes, large balls such as those that decide the lottery may be described just by classical physics so even if we use probability distributions, they're classical probability distributions. In principle, however, all the particles in the system obey the laws of quantum mechanics (which are only relevant if we want to be really accurate in our description of the balls) and in that case, it's true that the wave function evolves to a general superposition and the act of the measurement or its perception chooses one outcome or another, the "collapse". But the "collapse" isn't any material process affecting physical objects; it is the change of the knowledge leading one to replace the original probability distributions by conditional ones, conditioned by the freshly observed fact.
Again, as I mentioned at the beginning, it is possible – in these macroscopic, nearly classical, situations – to imagine that someone has "measured" or "perceived" the outcome before the citizen who is waiting for the lottery results at home so the "collapse" effectively took place before some people realized it. Whenever there is a possibility that the different microstates or outcomes may "reinterfere again", we must however be very careful to avoid any collapse up to the moment of an actual measurement, an irreversible perception of some classical information (usually accompanied by "decoherence").