Statistics Puzzle #1

We start by flipping n fair coins and remove any that land tails-up. We then flip the remaining coins again. What is the distribution of heads in this second round of coin tosses?

Hint: You may derive the answer using any method you like (eg using mathematics, writing a computer simulation or even trying the experiment in real life!)

Bonus: Starting from 1600 coins, we repeat the above process until we run out of coins. To the nearest integer, how many rounds do we need on average before this happens?