No, the same interference pattern observed in Young's double-slit experiment would not be obtained for two independent sources of light, such as the headlights of a distant car. The interference pattern relies on the coherent nature of the light waves emitted from a single source.
In Young's double-slit experiment, a single coherent light source illuminates the two slits, and the resulting interference pattern is formed due to the interference between the coherent waves from the two slits. Coherence implies that the waves maintain a constant phase relationship with each other over time.
If two independent sources of light, such as the headlights of a distant car, are used to illuminate the double slits, the light waves emitted from these sources would not be coherent with each other. The phases of the waves would be unrelated, and the interference pattern would not form as observed in the case of a single coherent source. Instead, one would observe two separate sets of diffraction patterns corresponding to the individual sources.
Therefore, the interference pattern in Young's double-slit experiment is a distinct characteristic of coherent light from a single source and would not be replicated by independent light sources.