Serendipity

Okay just woke up and checked my fragment list in Final Fantasy XIII-2 (FFXIII-2). I got the Lucky Coin fragment! For people having trouble with this fragment, here's what I did.

My walkthrough for Serendipitous PSN trophy and Lucky Coin fragment   

1. These steps are only valid if you have a fully developed Silver Chocobo with all chocobo racing abilities (mine didn't have Free Spirit but that didn't seem t affect its performance) available for Chocobo Racing and are sufficiently knowledgeable in the techniques used to win races (can trigger Sprinting Start regularly, know the different chocobo racing strategies etc.).
2. You can start with 0 gil in Chocobo Racing as races use Racing Points (RP) to enter, not casino coins or gil.
3. The first four races that you enter in your chocobo's career should be graded or secret in order to unlock the Fal'cie class races quickly. Fal'cie class races have the highest payout.
4. After that, only join races which have a payout of 320 or 480 coins for first. Also keep an eye out for Dahaka Stakes races which have a Collector's Catalogue as the first place prize. You can repeatedly farm these Collector's Catalgoues and leave with as many as 10 in one career. That's a whopping 110000 gil if you sell them to Chocolina.
5. You can attempt races that have interesting prizes such as adornments, weapons and accessories if you want. This only minimally affects your total winnings per career.
6. You can bet up to 100 casino coins on your chocobo per race to win a net gain of 120 casino coins if your chocobo wins first place. WARNING: DO NOT BET IN RACES THAT HAVE A RIVAL CHOCOBO WITH * RATING. Betting will buff all rival chocobos' stats significantly and reduce your chances of winning. Only bet if you are attempting any race lower than Weapon class.
7. You do not need Chocobull or Chocoboost. They are a waste of casino coins. If you win them as prizes though, feel free to use them in Fal'cie class races.
8. You should use Shuffle only if you are purely grinding for gold as it is not cheap in terms of casino coins. Use Shuffle to find Dahaka Stakes races quickly to farm Collector's Catalogues.
9. If you deplete your chocobo's RP, simply retrieve it to retire it and register it again to start a new career. Note: you have to repeat step #3 to u'nlock all race classes again everytime you start a new career.
10. If you follow these steps, you should obtain the Serendipitous PSN trophy in 2 careers. You should be able to make 6000 casino coins on average per career. There is a lot of misguided information out there. Serendipitous trophy requires you to win 10000 casino coins from Serendipity ??? AF. This includes the slot machines AND chocobo races over any number of careers or sessions (cumulative). Chocobo racing is by far the safer option of the two.
11. Once you have made about 20000 casino coins (or you can just buy them using the gil you made from selling Collector's Catalogues; remember, casino coins that were bought with gil or redeemed with tickets do not count towards the 10000 casino coins needed for the Serendipitous trophy), you are ready to try for the Lucky Coin fragment. To obtain this fragment, you are required to win at least 7777 casino coins from the slot machine. Only casino coins won from the slot machine are counted towards the total needed (NOT BOUGHT WITH GIL, TICKETS OR WON FROM CHOCOBO RACING). You can also amass this total from any number of slot machine sessions as it is cumulative.
12. Ask the attendant near the slot machine about the slot machine's mood.
13. If she says "It's like a hot summer's day!", proceed to step #12. If she says anything else, return to the Historia Crux and reenter Serendipity ??? AF to reset the slot machine's mood until she says "It's like a hot summer's day!".
14. Play the slot machine. Use a rubber band or cellophane tape to tape down the L1 button of your controller to leave the slot machine on Auto-play. 
15. Turn off your televison screen and leave your PS3 on for half a day. The most efficient time to do this is when you're going to sleep at night so you don't lose any precious hours.
16. Check your PS3 in the morning. Look at your fragment list. If you have the Lucky Coin fragment in the Serendipity ??? AF location list, congratulations! If not, repeat steps #12 to #15 but you can use less starting casino coins because you would have earned at least a decent amount of casino coins in your first attempt. 

To give you perspective of what to expect, this is my experience. I had a fully developed level 45 Silver Chocobo with all abilities except Free Spirit; 500000~ gil from finishing the story, collecting 157 fragments and defeating Raspatil, Ochu, Yomi and Long Gui multiple times; and 2000~ casino coins from redeemed tickets. I amassed 22000~ casino coins from 3 full careers in chocobo racing and sold 8 Collector's Catalogues I won as prizes for 88000 gil. I bought an additional 5000 casino coins using 50000 gil (1 casino coin costs 10 gil; that's why it's advisable to not buy casino coins and earn them through chocobo racing).

I started the slot machine with 27000~ casino coins in the mood "It's like a hot summer's day!" at 9PM last night and left it on overnight. I checked it this morning at 8AM, which means that I left it on for 11 hours. The slot machine had stopped Auto-Playing, I had 4 casino coins remaining (one play requires 5 casino coins) and 22 triple seven jackpots. Each triple seven jackpot bonus gives 250 casino coins and sends you into Victory Mode (which uses 10 casino coins per play). Each triple seven jackpot bonus in Victory Mode sends you into Super Victory Mode (which uses 100 casino coins per play). By calculation, 22 triple seven jackpot bonuses indicates that I won 8250 casino coins from triple seven jackpot bonuses alone without counting the other bonuses along the way such as triple chocobo bonus, triple cactuar bonus etc. I am unaware of the mechanics of triple microchu jackpot bonus as I have never gotten any.

I was monitoring the slot machine on Auto-Play for the first hour that I left it on. When I won two consecutive triple seven jackpot bonuses, I did not get the cutscene that gave me the Lucky Coin fragment. This disproves the myth that some people have been spreading that the Lucky Coin fragment can be obtained by winning a triple seven jackpot bonus. The ONLY way to get the Lucky Coin fragment is by winning at least 7777 casino coins from the slot machine. On the play that pushes you past the 7777 casino oin mark, you will get the cutscene. Sadly, I was asleep when it happened. Oh well, there's always Youtube.