# What are the chances of shooting the moon in Hearts?

I'm posting this again here from boardgames.SE because it was suggested there that this is the more appropriate question to ask it.

So, that said, here's my question: given a fair shuffle, that all players have perfect information of everyone else's hands, and are all playing perfect games, then what are the chances of getting a hand that is guaranteed to shoot the moon in Hearts?

I suspect that there is some probability that exists, because if we have a hand with all 13 clubs, or an Ace of clubs and the 3 through Ace of heart cards, there is no possible configuration where we'll NOT shoot the moon.

Whereas, if we get a full suit of diamonds, hearts, or spades, there is no configuration where we'll ever be able to shoot the moon.

All hands in between could then either be a shoot to the moon hand or not. But what fraction of them are?

If you aren't familiar with the game but would like to be, I have a link here. But basically, here are the rules:

1. A deck of 52 cards is distributed to 4 players, so each has 13 cards

2. The first person to put down a card is the one with the 2 of clubs, who will then put down the 2 of clubs

3. All other players must put down cards with the same suit as the first card (for the first round, it mus be a clubs)

4. In the case that a player is void of a suit (0 cards on hand of the suit) which the first player leads (first player to put down the card -- in the first round, this is a clubs), they may put down any card they want

5. The one who gets the trick (collection of cards placed down in a round) is the one with the highest value card, if it is the same suit as the first player's lead card

6. For every heart card one receives, one gets 1 point. The Queen of Spades is worth 13 points

7. The aim of the goal is to get the lowest possible score before someone reaches 100 points.

8. A shot to the moon is when all 13 hearts and the Queen of Spades is won by only a single person, in which case all the rest get 26 points and the shooter's score does not change.

So, basically, the Microsoft version of hearts.