Let $T_x$ denote the first time a symmetric random walk visits $x$. (The random walk starts at $0$.) Find $\mathsf E(\min(T_a, T_{-b}))$ where $a, b > 0$.
Hint: we computed $P(T_a < T_{-b})$ previously by conditioning on the first step. A similar approach might work here as well.
I have been looking at the expected duration of the Gambler's Ruin problem as an analogy but I haven't gotten very far this way. Any help would be appreciated.
Observe that by the law of total expectation, $$E[\min\big\{T_a, T_{-b}\big\}] = E[T_a | T_a < T_{-b}]P(T_a < T_{-b}) + E[T_{-b} | T_{-b} < T_a]P(T_{-b} < T_a)$$