Yes, in your example they would trace the coins and find out who you are. All transactions can be tracked because everything is publicly logged. Check blockexplorer
To make "anonymous" transactions you must launder bitcoins. There are services
that already do that, but I don't know if they are very used or not. For this to work, you must have a lot of people involved.
If a service like this works, the idea is simple. I'm going to quote David Schwartz from a similar question at Bitcoin Stack Exchange
(this will go public on tuesday):
Say I received 10 bitcoins on a Bitcoin address I publicly advertise for donations. Anyone looking at the block chain can put that address in a search engine and fine me. Now say I want to use those 10 bitcoins to buy drugs. If the drug dealer's Bitcoins are traced, they'll point right back to me.
Now I can create a lot of accounts if I want. And I can pass the 10 Bitcoins I received through those accounts and then to the drug dealer. But likely because those accounts aren't used anywhere else, the trail would still lead right from the drug dealer to me.
Now suppose I put my 10 Bitcoins in an account created just for me by a Bitcoin laundering service. And you and Fred do the same thing. Then the service sends my coins to Fred and your coins to me. Then I spend the coins I got on drugs. Now the trail looks exactly the same as in the non-laundering example, except the trail leads straight to you instead of me -- a pretty good deal for me, not so much for you.
What happens in reality is that those 10 Bitcoins are split among lots of people lots of times, which creates a big mess. It wouldn't be possible for someone from the outside to find out where the money really came from.