Let's say $M$ people play the game:
Player $i$ has $n_i$ cards with his name where $1\le i\le M$
All cards put in the hat. Then a card is randomly drawn from the hat. The owner of the card is the winner. All winner cards are removed from the hat. Everything is repeating in the next round, but there are no last round winner cards.
So, the question is
What is the probability that a player $i$ wins a game of $K$ rounds?
Note:
I know how to solve the problem using conditional probability: for an example, there are 3 players: $n_1=3, n_2=2, n_3=5$ and 2 rounds. The probability that the first player wins $P_1=P(A) + P(B)$ where $A$ - the first and the second players are winners, $B$ - the first and the third players are winners.
Let's say $Q_p^r$ means the fact that the $p$ player's card is drawn at the $r$ round
Then $P(A)=P(Q_1^1)\cdot P(Q_2^2 | Q_1^1) + P(Q_2^1)\cdot P(Q_1^2 | Q_2^1) = \frac{3}{10}\cdot\frac{2}{7} + \frac{1}{5}\cdot\frac{3}{8}\approx0.1607$
$P(B)=P(Q_1^1)\cdot P(Q_3^2 | Q_1^1) + P(Q_3^1)\cdot P(Q_1^2 | Q_3^1) = \frac{3}{10}\cdot\frac{5}{7} + \frac{1}{2}\cdot\frac{3}{5}\approx0.5143$
So, $P_1 = P(A) + P(B) \approx0.6749$
It is easy in this case, but how to solve the problem in general? What if $M$ and $K$ are much larger? Maybe there is a pretty elegant solution?