What are the chances of shooting the moon in Hearts?
It's certainly possible to calculate which hands can Shoot the Moon with 100% probability, but it's not going to be easy.

There are 635,013,559,600 different starting hands of 13 cards from a 52 card deck.

We know that if we are dealt all 13 cards of one suit, there's 4 hands out of  635,013,559,600 that will shoot the moon ;)

If we have 12 cards of one suit and one ace in another suit, there are 13 different ways to get 12 cards of one suit, times 3 different other ace cards in other suits, times 4 different suits so: 13*3*4 = 156 winning hands.

Of course if we have an ace and king of the other suit and the rest one suit, it's also a winning hand — wait it has to be an Ace-King in the long suit too because we can't risk an opponent having a king plus another card in our long suit.

Well, this is getting difficult soon. Thankfully, Bridge players have already figured this out for us. :) They have a concept called the Losing-Trick Count, a method of hand evaluation for counting the number of losing tricks held by a partnership. To "shoot the moon" we need a Losing-Trick Count of Zero. 

According to this table I found on Wikipedia, there are 4,245,032 hands with a losing trick count of zero. As a check I summed up the number of hands in that table and it equals 635,013,559,600 exactly. Whew! Sounds right!

Therefore, your probability of Shooting the Moon with 100% accuracy is at a bare minimum .000668% percent.

HOWEVER… Trick points are scored differently in Hearts as compared to Bridge. You can lose tricks but still take all the hearts and the queen of spades. So Ace-King of Spades + AKQJT98 of Hearts plus 6 other losers is a winning hand. I expect about half of the 90,206,044 hands with a LTC of 1 are shoot the moon winners since if the losing trick is a club or diamond, at that point it doesn't matter. (Presuming the winning tricks have forced the Queen and hearts. Things get murky.)

So, I expect the number of perfect shoot the moon hands is significantly more than the .000668% percent of Bridge hands that are perfect. You can add at least .0071%, and then a bit more as you start chipping away at those losing tricks that no longer matter.

My brain is starting to hurt now.

I think I'll go lie down and hopefully this gives you an idea of how to start to get the answer. It's important to realize that the reason shooting the moon isn't as rare as you'd expect is because even a marginal hand has a decent chance of shooting the moon, for your opponents don't have perfect knowledge. If an opponent has a king-queen in one suit and a king-10 in the other, which king do they keep to stop your attempt?

And of course, at first most opponents don't expect players to go for the moon, and are trying to get rid of trick taking cards when they can. That's what makes Hearts so interesting in my view, the psychology of being wary of those greedy moon targetters. ;)