Why do we need an earth wire? Apologies if this question has already been asked before.
In this video and other sources, it says that the ground/earth wire is connected to the outside metal casing of an electrical appliance in order to create a low-resistance path back to the live wire in case of a fault. However, if the ground wire wasn't there, how would current be able to flow into the person and cause an electric shock if there was no "other end of the circuit"?

The best explanation I can find so far is that the live wire will "electrify" the casing or to make it become "live", but I'm not sure if this is possible even if the circuit is not completed. Is it possible for electrons to flow along the casing even if there is nowhere for them to go?
Edit: Sorry, my question wasn't phrased very well. If the neutral wire is also connected to the ground, how does the resistance of the circuit inside the machine compare to the ground? My assumption is that the resistance of the ground is higher, but I may be wrong on this.
 A: First, keep in mind that there is voltage between the "live" wire, more properly called the ungrounded circuit conductor, and the earth due to the fact that the neutral conductor is bonded to earth along with the earthing wire (more properly called the equipment grounding conductor) at the circuit panel.
Without an equipment grounding conductor connected to the metal housing, the fault shown in your diagram will result in the entire housing being at line voltage with respect to the earth. If the person touches the housing and is also touching earth the person completes the circuit back to the supply. See the first figure below.
With the equipment grounding conductor connected to the metal housing, the fault current flows through the housing and equipment grounding conductor (which has a much lower impedance than the person) back to the supply, effectively bypassing the person, and tripping the circuit breaker (not shown) terminating the fault. See the second diagram below. In effect, grounding the housing lowers the voltage between the housing (and person) and the earth.
Hope this helps.


A: The ground the person is standing on would be the return path. The Earth is an effectively infinite sink for current, at 0 V.
A: It's worth noting that, for example, the metallic wire conduits and water pipes in your house are grounded, so it's by no means obvious what the lowest-resistance path to ground might be. The housing of my toaster oven is made of metal, so what's the closest path to ground? The faucet? The refrigerator? My cabinet handles? The metallic body of my conventional oven which is a few feet away?
Imagine a fault occurring in some appliance that you're touching, and ask yourself how certain you are that there is absolutely nothing nearby which is connected to a pipe or a conduit or a large appliance or wet soil or [.....]. Are you sure enough to risk your life or your families lives, when the alternative is to add one extra wire to the appliance which would save you in the event of a (even highly unlikely) mistake?
As a toy$^\ddagger$ example, imagine that somehow a fault develops in your ungrounded dishwasher which connects the nice, aesthetically-pleasing brushed steel panel on the front of the appliance to the hot supply wire connected to the outlet under the sink. If you're standing next to the dishwasher and not touching anything, then your resistance to ground is probably quite high, so you should be safe even if you grab the electrified handle. But that's not how people use their appliances at home.

*

*Have you ever been loading your dishwasher and touched the door at the same time as your metallic sink basin?  Zap. You forgot that the sink is connected to your water pipes, which are grounded.

*Have you ever touched the dishwasher at the same time as you were opening a cabinet? Zap. You forgot that the cabinet handles are metallic, and that they're screwed into brackets which run along the side of the cabinet doors, and that there's an electrical conduit box mounted on one of those rails on the other side of the kitchen.

So on and so forth. The point is that if your resistance to ground is extremely high, then you won't receive a shock by touching your electrified dishwasher ... but there are a near-infinity of ways that an electrical ground may be far nearer than you think, either by accident or design.
If your dishwasher were grounded properly, then the instant your fault developed it would draw enough current to trip your circuit breaker. Annoyed, you'd go to the breaker and reset the switch and it would instantly trip again. You would then call an electrician or a contractor who would immediately know that something is shorted to ground, and who would fix or replace your appliance. Inconvenience seems a fair trade, considering the alternative.

$^\ddagger$This is a toy example because I have no knowledge of plumbing or electrical codes and standard practices. I'm inventing a scenario which may or may not be realistic from the perspective of an industry professional, but I am a mere physicist so I am treating home wiring as a spherical cow in a vacuum.
