In the double-slit experiment with single electrons, the mathematical model of quantum mechanics predicts the interference pattern precisely, not just on average. Quantum mechanics describes the behavior of particles, such as electrons, in terms of wave functions, which are mathematical objects that contain information about the probability distribution of a particle's properties.
When a single electron is sent through the double-slit apparatus, its wave function undergoes a phenomenon known as interference. The wave function interacts with itself as it passes through both slits and creates an interference pattern on the screen behind the slits. This interference pattern consists of alternating bright and dark regions, indicating regions of constructive and destructive interference, respectively.
The predictions of quantum mechanics for the interference pattern are precise and deterministic. The mathematics of quantum mechanics provides a way to calculate the probability distribution for the electron's position on the screen, and this distribution matches the observed interference pattern remarkably well.
It's important to note that while the interference pattern is deterministic, the specific outcome of where an individual electron will land on the screen cannot be predicted precisely. Quantum mechanics only provides probabilities for the outcomes of measurements, and it is inherently probabilistic in nature. However, when many electrons are sent through the apparatus, the interference pattern emerges clearly and matches the predictions of quantum mechanics.