I gather from the name that they're bullets with a hollow tip, but I can't figure out why that would be beneficial.
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When shot, the bullet takes air resistance. The air pushes the hollow point on the bullet and sort of flattens it in midair. Then there's stronger hollow point bullets which expand only when they come into contact with the target. The whole point of this is to make the bullet bigger. The bigger the bullet, the more damage it does. They can't fit giant bullets in pistols without making them heavy and inconvenient. So they make bullets that get bigger when shot. Also, pointy full metal jacket bullets go through targets. Sometimes, at high speeds, these bullets will go through the target with small holes, and the target won't even feel it at first. This let's the target run away. The hollow point has bigger force which knocks the target harder. This is very useful for hunting fast escape animals. The bullet isn't hollow on the inside, it's just got an inner bump on its front.
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Perhaps the most popular reason for choosing hollow-point rounds is the expansion factor. Hollow-points are simply less dense, because they are partially hollow for a given bullet diameter, which allows them to mushroom when impacting a target. This mushroom effect increases the diameter of the round, which in turn, creates a larger wound in the animal, resulting in a quicker and more humane death. This mushroom effect is also desirable of a bullet that must travel long distances to reach its target. Even after losing energy downrange, a hollow point will still open up on impact.
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You are correct that they have an opening (hole) in the nose. Because being partly hollow, they will expand or "mushroom", and sometimes shatter, on impact. This creates a larger wound. However since there is less material, thus lighter weight, they don't carry quite as much energy as a solid bullet.
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A hollow-point round is designed to expand upon contact, thereby doing extensive damage to body tissue. Solid bullets may do less damage since they penetrate without expanding, and pass right through the tissue.