I posted below that no probabilities are needed in quantum mechanics. A reader asks whether the Double-slit experiment proves randomness in quantum mechanics.
I don't see how any physical experiment could prove randomness, or directly observe probabilities. The double-slit experiment was performed and understood by Young in 1803, and no one thought then that it implied anything about probability or randomness.
I think that the relation is as follows. If you assume that light is composed of classical (non-quantum) photons, then the photons go thru one slit or the other, with some probability for each. The diffraction pattern can also be interpreted as a probability distribution.
You get similar results if you collect data on coin tosses. It does not say anything about whether the coin tosses are deterministic or probabilistic.
To get the quantum probabilities, we had to assume classical particles, which is clearly an invalid assumption. Quantum mechanics is emphatic that light does not consist of classical particles. Sometimes quantum mechanics textbooks say that light is composed of particles, but then they are funny particles that can be in two places at once. So we cannot say that the light particle goes thru one slit or the other.