Birthday Paradox

Someone told me that if there are 20 people in a room, there's a 50/50 chance that two of them will have the same birthday. How can that be? "When you put 20 people in a room, however, the thing that changes is the fact that each of the 20 people is now asking each of the other 19 people about their birthdays. Each individual person only has a small (less than 5%) chance of success, but each person is trying it 19 times. That increases the probability dramatically." A use for this is in crypto which has a fairly impractical problem with key exchange.