# Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such that it has four cards and repeat removing. I win the game if I can remove all 52 cards from the board and lose otherwise, i.e. when all cards are from different suit. What is the probability to win this game?

I guess we need some kind of generating polynomial but I'm not sure how to solve that kind of problems.

• I think you can't win the game, removing all $52$ cards and clearing the table. The max you can remove is $51$ (so you'll have one card left on the table at the end). Simulation seems to bear this out, though I can't right now figure out a complete proof. – ShreevatsaR Jan 7 '14 at 17:42
• It's certainly possible to win, for example with a deck configuration with 11 hearts followed by 1 spade, two hearts, 12 spades, and then similarly with clubs and diamonds. – Jonas Granholm Apr 9 '14 at 16:18

$$\frac{4611922675628644134029}{721031421579150441387600}\approx0.0063963\;.$$