The size of the sensor isn't as important as people think it is. What really matters is the diameter of the aperture (the absolute diameter, not the f number). Consider a cone of light for a given angle of view hitting a small sensor close to the aperture and a large sensor further from the aperture:
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If you compare typical shooting apertures for DLSRs and camera phones, they're not radically different. Say you are shooting a 50mm lens at f8 on a DSLR. That's an aperture of 6.25mm. A typical smartphone camera will have an aperture of around 3-4mm. In this scenario, then, the DLSR is getting about 3 times more light (or ~1.5 stops).
Of course you can use much wider apertures on DSLRs, but their use is more limited given the shallow depth of field that results. If you're shooting e.g. landscapes, then you're probably not going to use apertures much wider than f8 anyway.