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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.

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