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What is the chance of Earth being hit by a comet or asteroid?
Not much in our lifetimes -- perhaps 1 in 10,000 -- but over thousands or millions of years, major impacts become pretty likely. Ancient craters on Earth's surface prove that large objects have hit Earth in the past, and there's no reason to think this won't continue in the future.
The chance of an impact depends on the size of the object: the bigger the comet or asteroid, the smaller the chance, since there are many more small objects out there than large ones. Tons of debris -- much of it in pieces smaller than grains of sand -- strike Earth's atmosphere and burn up every day. These are the "shooting stars" commonly seen at night. Some larger rocks survive their fiery descent to the surface; you can see some of these "meteorites" displayed in museums. The truly dangerous objects, those large enough to cause regional or global catastrophe when they hit, may appear once every few hundred thousand years. Therefore, the chance that such an object will hit us in any given year is roughly 1 in 300,000 -- nothing to lose sleep over.
Many scientists believe an extremely large asteroid (about six miles in diameter) struck Earth 65 million years ago near the present-day Yucatan peninsula of Mexico. The impact caused catastrophic conditions across the entire planet, including thick clouds of dust and ash that caused global temperatures to plummet, causing the extinction of the dinosaurs and much of the rest of the life on Earth.
The path Earth follows in its orbit around the Sun is littered with untold pieces of debris. Unlike the dinosaurs, we have the means to find the largest of these "Near-Earth Objects" (NEOs) and calculate their orbits, to see if they might ever come close to us. Currently, several different telescopes routinely and automatically scan the sky for them.