This is actually deceptively simple. A full set includes every card of that type in the game. If you possess a card in a specific set, then nobody else can complete that set. If you possess cards from every set, then no set can be completed without trading with you.
The flip side is that if there is any set that you lack cards from, then that set can ...
This generalization of Tic-Tac-Toe is called m,n,k-game. (the goal is to get k in a row on a (m,n) board).
Some known bounds: (source wikipedia)
(5,5,4) is a draw.
(6,6,5) is a draw.
(7,7,5) and (8,8,5) are draws.
(15,15,5) is a win.
(9,6,6) and (7,7,6) are both draws via pairings.
When the goal is 9 or larger (k>=9) the second player can force a ...
Suppose you're playing a three player game with wheat, barley, and corn. For one of the trading players to win with wheat, the two trading players must have nine wheat between them, which is the same as the shut out player not getting any wheat.
To get a hand, we choose 9 cards out of 27. There are C(27,9) ways of doing that, where C is the binomial ...
(I'm mostly ignoring Bull and Bear rules here, but I think they can be added in with a little fiddling.)
If you just want to know whether it's possible, then it's not a question of probability but of showing that there is at least one arrangement of cards that would allow a player to win without trading. And it's pretty easy to see that this must exist, ...
There are several generalizations to tic-tac-toe. The most natural one (imho), is the m,n,k-game, which is the game of k-in-a-row played on an (m,n) board.
In (n,n,n) games, for n>2, the second player can force a draw. See m,n,k-game for many other results.