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The probability " wave " can be said to " pass through space " because the probability values that one can compute from its mathematical representation are dependent on time.
One cannot speak of the location of any particle such as a photon between the time it is emitted and the time it is detected simply because in order to say that something is located somewhere at a certain time one has to detect it.
The requirement for the eventual appearance of an interference pattern is that particles be emitted, and that there be a screen with at least two distinct paths for the particle to take from the emitter to the detection screen.
Experiments observe nothing whatsoever between the time of emission of the particle and its arrival at the detection screen.
If a ray tracing is then made as if a light wave ( as understood in classical physics ) is wide enough to take both paths, then that ray tracing will accurately predict the appearance of maxima and minima on the detector screen when many particles pass through the apparatus and gradually " paint " the expected interference pattern.

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