Serendipitous Achievement in Final Fantasy XIII-2
Amassed a fortune in casino coins.
How to unlock Serendipitous
You will need to have winnings of 10,000 Casino Coins for this achievement to unlock. This requires that you be UP 10k, so even if you've won 10k, but spent 2k doing it, that means you're at 8k worth of winnings. You can win the coins via the races or the slot machines, but I would suggest using the slot machines to do this, since you need to spend some time there to earn the "Lucky Coin" fragment anyway (by earning a total of 7,777 Casino Coins at the slot machines) and the chocobo races can be extremely tedious.
The easiest way to do this would be to save your game, buy a large amount of coins from the exchange, then rubber band your to have the machine auto-play. Check back later and see if you got your fragment or this achievement. If you've run out of coins without getting either, reload your save and try again later.
Some extra slot machine tips can be found here. Note that the achievement will only unlock once you exit the slot machine. If you're having absolutely no luck with the slot machines, you can do the chocobo races. A good guide for doing them quickly can be found here.