In the double-slit experiment with single electrons, if the lateral position is measured very precisely, it will indeed provide information about which slit the electron passed through. However, the act of measuring the position with high precision will disturb the electron's momentum, leading to a loss of interference pattern on the screen.
The mathematical model of the experiment in quantum mechanics can successfully predict the probability distribution of where the electron is likely to strike the screen, given the knowledge of which slit it passed through. This probability distribution can be calculated using the principles of quantum mechanics and the wavefunction of the electron.
When the position of the electron is precisely measured, the wavefunction of the electron collapses to a localized state centered around the measured position. As a result, the interference pattern disappears, and the electron behaves more like a classical particle with a definite trajectory. The probability distribution predicted by the mathematical model will then correspond to a particle-like pattern on the screen, with a higher probability of detecting the electron at the corresponding measured position.
It's important to note that the loss of interference and the transition from a wave-like behavior to a particle-like behavior when measuring the position of the electron is a fundamental aspect of quantum mechanics known as the wave-particle duality.