Choose Your Order

You and your friend are playing Mario Kart. Each turn, either you or your friend gets to select the map. If you select the map, you have probability $p_1$ of winning the turn. If your friend selects the map, you have probability $p_2$ of winning the turn. Consider the following two possible conventions for map selection: $\textbf{Method A:}$ Alternating every turn $\textbf{Method B:}$ The player selecting continues to select until they lose You get to select the map of the first turn. Each time someone wins, they receive a point. The first person to reach $n$ points wins the tournament. Assuming you know both $p_1$ and $p_2$ , under an optimal strategy, which convention would you choose when $n = 10$ to maximize your probability of winning?