Why does the photon strike at one or another place on the tape? I am more than happy in accepting the wave nature of light and the fact that each of photons has probabilistic distribution "coded" within itself. But what is about the particle's part and the locality principle?
I can understand the essences of the quantum mechanic world and the fact that it was built upon the probability theory and acceptance of existence of truly random variables. But what is about hidden variables? Why is the concept of truly random variable so comfortable for physicists?
 A: First I will answer the title question. I will assume you are talking about the double slit experiment and photons being shot one at a time.
As per quantum mechanics, it has a dual nature, sometimes it manifests particle properties and sometimes wave properties. Whether you see the particle or wave properties depends on the measurement, photons manifest this duality, showing the frequency associated with the whole ensemble of EM wave. The classical EM wave is made of a herd of photons, and the two models, classical and QM fit perfectly.
Your question is basically if we know the direction of the emitted photon (its direction after interacting with the slits) upfront. The answer is no. In QM it is the probability distribution that is determined, and the individual photons will be in line with that distribution. The direction of the photon is random, but it will obey energy and momentum conservation and as the experiment is repeated many times, the photons will have to follow the probability distribution.
Now your question about the direction and the hidden variables arises because light is not a particle or a wave, it is a quantum field, the photon field in this case, and this field can interact with the atoms in the slits and the screen in a wave or particle way.
The light source interacts with the photon field and the energy propagates outwards in a spherical shape in every direction. As the energy of the photon field interacts with the screen, you see this as a dot. That is when the photon is localized, but not before that.
Before that, the photon is in a superposition of all locations in space and time (as it propagates in all direction outwards from the light source). The interaction with the screen collapses the wavefunction and that is when we localize the photon on the screen. Before that, we cannot know the direction of the photon nor its position, since as a wave it passes through both slits before it interacts with the screen and leaving a dot on the screen.
