11. Team A and Team B each send out 7 players to participate in a Go tournament in a predetermined order. Both sides first send out Player 1 to compete, the loser is eliminated, and the winner then competes with the losing side's Player 2... until one side's players are all eliminated, and the other side wins. Find the probability that Team A has 6 players eliminated and finally wins.
11. Party A is eliminated 6 players, and finally wins, corresponding to 13 eliminated players $a_{1}$, $a_{2}$, $a_{3}$, ..., $a_{13}$, where $a_{13}$ is the 7th player of Party B, and $a_{1}$, $a_{2}$, $a_{3}$, ..., $a_{12}$ are six each from Party A and Party B, so there are $\mathrm{C}_{12}^{6}$ ways.

Each match process corresponds to one side taking 7 out of 13 eliminated players, so there are $2 \times$ $C_{13}^{7}$ possible match processes.
The probability that Party A is eliminated 6 players and finally wins $\mathrm{P}=\frac{\mathrm{C}_{12}^{6}}{2 \times \mathrm{C}_{13}^{7}}=\frac{7}{26}$.