Unless it is cut off in an accident, a finger will not normally fall off on its own.
There are several medical conditions that can affect the arterial blood flow to extremities (arms, legs, fingers, toes) and eventually cause their loss or require their removal. These include complications from Diabetes, or tissue destruction (necrosis) as from gangrene infection. One of the most well-known examples of damage to fingers, toes, noses, and ears is from frostbite, or hypothermia, in the polar regions or atop high mountains.
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