How likely are you to lose because Fool's Landing sinks into the abyss?
A friendly warning: This is a looong answer.
  
  I've calculated the chances of Fools' Landing being lost immediately after the first turn with all players playing to prevent it and a random tile setup.


A few variables:


n = number of players  
w = water level after rising once (2 for Novice, 3 for everything else)


Special abilities/actions

First, of all, there are a couple of special abilities and cards that make it impossible to fail:


If the first character is the Pilot (1/6), they start at FL and can always shore it up, or  
If anyone starts with a Helicopter Lift (there are 3) or a Sandbag (2) in their hand (1-(20 C (2n))/(25 C (2n))), they can shore up or move the first player to shore up (it's OK to ignore WR cards in the initial deal)


1/6 + 1-(20 C (2n))/(25 C (2n))

i.e. (1 - (1/6 + 1-(20 C (2n))/(25 C (2n)))) chance that the following permutations can end the game.

Floods, Waters Rise and More Floods

What the characters do on their turns is tricky, so we'll deal with all the card probabilities together first:


6 areas are flooded to start. To immediately fail, FL must be one of them (6/24)  
At the end of the first player's turn, regardless of what they did, a fail requires a WR card be drawn from the remaining deck. They draw two cards. The first has a 3/(28-2n) chance of being WR. If it's not ((25-2n)/(28-2n)), the second card has a 3/(27-2n) chance of being WR. After some balancing, that becomes (156-12n)/((28-2n)(27-2n)) chance that a WR card is drawn on the first turn  
If it is, the (6-card) discard pile is shuffled into the deck and there is a w/24 chance that FL comes out again.


So, the odds that the cards are against you and only a perfect first turn can prevent a fail is:

6/24 * w/24 * (156-12n)/((28-2n)(27-2n)) = (w(156-12n))/(96(28-2n)(27-2n))

The First Turn: Explorer Edition

This is where it starts to get very hairy, so bear with me. If the first player is the Explorer, they can move diagonally, so that makes everything a bit more reachable. There is a 1/5 chance that the first player is the Explorer (because we've already taken into account that they're not the Pilot). To fail as the Explorer, you have to be more than three spaces away from FL including diagonally.


From the outside 8 spaces (hereafter, "outside"), there are six you cannot reach. The odds of placing FL in one of the 6 unreachable spaces is 6/23. Therefore, there is a 8/24 * 6/23 = 48/552 chance you fail for this starting position  
If you start in a corner of the inner square ("corner"), there are four spaces you cannot reach as an Explorer (the opposite outsides). That's a 4/24 * 4/23 = 16/552 chance of failing  
If you start in a space that is not in the centre four, "outside" or in a "corner" (which I'll refer to as "inside" for clarity), there are two unreachable spaces. That provides 8/24 * 2/23 = 16/552  
Finally, your first option could be one of the central spaces (the centre four). If it is, congratulations! You can reach any space!


So, the odds of being the Explorer and being unable to shore FL on your first turn, based on all tiles being placed randomly is:

(48+16+16)/552 * 1/5 = 16/552

The First Turn II: Everyone Else Edition

But what if you're not the Explorer?! Let's look at all the potential starting spaces again (groan...) given you're one of the other non-Pilot characters (4/5):


There's a 80/552 chance that FL is unreachable from outside  
There's a 40/552 chance that FL is unreachable from a corner  
There's a 72/552 chance that FL is unreachable from inside  
There's a 12/552 chance that FL is unreachable from the centre.


So, the odds of being a non-Explorer, non-Pilot and being unable to shore FL on your first turn is:

(80+40+72+12)/552 * 4/5 = 816/2760

Therefore, the odds of being unable to shore up FL (given that you're not the Pilot) is:

16/552 + 816/2760 = 896/2760

Phew!

The Grand Conclusion, Or TL;DR

So, we now take all three sections' results and multiply them together:


The odds that you're not the Pilot and no one has cards that can help you = (1 - (1/6 + 1-(20 C (2n))/(25 C (2n))))  
The odds that you all the flooding will go against you, and you'll draw at least one WR = (w(156-12n))/(96(28-2n)(27-2n))  
The odds that you won't be able to reach and shore up FL under normal circumstances = 896/2760


Given that w=3 (which it is for most setups), the odds of failing on the first turn by number of players is:


2 players: approximately 0.0768%  
3 players: approximately 0.0483%  
4 players: approximately 0.0288%