Probability by the Bucketful

Suppose you have a large number of buckets and an equal number of balls. You randomly pick a bucket to put each ball in one at a time. When you’re done, about how what proportion of buckets will be empty?

One line of reasoning says that since you have as many balls as buckets, each bucket gets one ball on average, so nearly all the buckets get a ball.

Another line of reasoning says that randomness is clumpier than you think. Some buckets will have several balls. Maybe most of the balls will end up buckets with more than one ball, and so nearly all the buckets will be empty.

Is either extreme correct or is the answer in the middle? Does the answer depend on the number n of buckets and balls? (If you look at the cases n = 1 and 2, obviously the answer depends on n. But how much does it depend on n if n is large?) Hint: There is a fairly simple solution.

What applications can you imagine for the result?