Okay, so I came across an interesting probability question that has left me stumped to come to a solid conclusion.
Assume that you are playing a card game with two opponents, $A$ and $B$. Assume that the probability of winning against $A$ is greater than winning against $B$. Probabilities of winning against $A$ or $B$ are independent. You will play these opponents one at a time in alternating order. You receive a payout only if you are able to win two games in a row out of a three game series.
Given the choice, would you choose to pick $A$ (i.e. Would play $ABA$) or $B$ first (Would play $B A B$)
My instinct tells me that you would be better of choosing $A$ first, but since any two wins in a row would involve beating $A$ and $B$, I am not convinced that it matters.
I am curious if there is a more mathematical explanation than my instinct.