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In the game of spades, a standard deck is shuffled then all the cards are dealt in a clockwise manner until each of the 4 players has 13 cards. The first play of the game is for each player to throw their lowest club (clubs are ordered from low to high: 2,3,4,...,Queen,King,Ace).

When all four lowest clubs are on the table, the player who threw the highest of those four cards wins the "trick" but if a player has no clubs, he or she must play a heart or a diamond, and that card has no chance of winning the trick. If a player has no clubs, no hearts, and no diamonds, then the player must play a spade, and will be guaranteed to win the trick.

I simulated this game by counting how many times a specific card won and I divided it by the number of tricks and I got a winning probability of 0 with the card 2 of Clubs, I got an approximate probability of 9.15 with 10 of Clubs, 3.59 with King of Clubs, and 11.98 with 9 Clubs but now I want to solve it mathematically. I think simulation is way too complicated because I have simulate 1,000,000 tricks or more to get closer to the true value.

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    $\begingroup$ What do you mean with "probability 9.15"? Is that in %? $\endgroup$
    – Arthur
    Commented Oct 30, 2013 at 13:45
  • $\begingroup$ When you don't have any other way to demonstrate something, simulation is a very good way. $\endgroup$
    – Claude Leibovici
    Commented Oct 30, 2013 at 15:21

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You haven't defined the rules completely. If all the hands are $4-3-3-3$, the middle five tricks will have no clubs or spades, so a heart or diamond must win. But I think simulation is the only reasonable approach here-there are too many different possibilities for hand work. If a card has a $10\%$ chance of winning a trick, the error in $10,000$ runs will be about $\sqrt {10,000\cdot 0.1 \cdot 0.9}=30$ or about $0.3\%$ and it doesn't seem even a million runs with error about $0.03\%$ should take very long. You can keep track of the chance for every card on one run through, so don't have to do a separate run for each card.

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