Which of these colours is least scattered by dust, fog, smoke?

The correct option is Red.

To understand why red light is the least scattered by particles like dust, fog, or smoke, we must look at the physics of light scattering, specifically Rayleigh scattering.
Rayleigh scattering describes the scattering of light by particles that are much smaller than the wavelength of the light itself. The intensity of the scattered light ($I$) is inversely proportional to the fourth power of its wavelength ($\lambda$):

$I\propto \frac{1}{{\lambda }^4}$

According to this relation, light with shorter wavelengths scatters much more than light with longer wavelengths.
In the visible spectrum, different colors have different wavelengths. The approximate wavelengths for the options provided are:
- Violet: ~400 nm (shortest wavelength, scatters the most)
- Blue: ~450 nm
- Yellow: ~580 nm
- Red: ~700 nm (longest wavelength, scatters the least)

Because red light has the longest wavelength among all the visible colors, it experiences the least amount of scattering by small particles in the air such as dust, smoke, and fog. This allows red light to travel longer distances through these scattering mediums without losing its intensity, which is why red is universally used for danger signals and tail lights.