Order of chess opponents to win two in a row

Alice will play three chess games, alternating between two opponents, Bob and Carl, and she claims a prize if she wins at least two games in succession. She may choose the order Bob–Carl–Bob (option 1) or Carl–Bob–Carl (option 2). Carl is the stronger player. Which option gives her the better chance of the prize? Answer $1$ or $2$.

Option 2. With win probabilities $b$ against Bob and $c<b$ against Carl, she needs to win the middle game and at least one of the outer games. Option 1 gives $c\,[1-(1-b)^2]=bc(2-b)$; option 2 gives $b\,[1-(1-c)^2]=bc(2-c)$, which is larger. Facing the stronger player twice is better because the middle game against the weaker one is the must-win.