Cables and circuits inside an appliance is insulated from the rest of the appliance so that the electricity will not flow from a live wire to the chassis or body of the appliance which the user may touch. But if something goes wrong with this insulation you could have a live wire touch the metal that the user is touching which is dangerous. This is why you need to run a ground wire from the parts that the user can touch to the ground to allow any stray current to flow through this wire instead of through the user.
A double insulated appliance however not only insulates the live wires inside it but also the parts that the user might touch. Most commonly today is to make the body of the appliance out of plastic. In this case even if the insulation on the wires and circuit boards may not be working properly the current can not get to the user. So no grounding is necisary.
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