Since this is tagged with “Puzzles and Trick Questions” it may be that I am missing something. But my answer would be [math]\Sigma_{m=0}^{n-1}p^{2n-1-m}(1-p)^m[/math] .

This follows the pattern of the best 2 out of 3 case where Alice has to win either two or three games – which happens in cases lww,wlw,wwl or www with probability [math]3p^2(1–p)+p^3=3p^2–2p^3[/math] (where the fact that the game may be stopped when she wins twice just corresponds to the fact that [math]pp(1-p)+ppp=p^2[/math] , and the same answer is obtained by taking the complement of the cases where Bob wins either 2 or 3 games).