Consider a state $i$ from some state space $A$. First passage time to state $i$ is the random variable $T_i$ defined by
$T_i(\omega) = inf$ { $n \geq 1: X_n(\omega) = i$ }.
Does this means that first passage time is the minimum number of steps required by a random variable $X_n$ to attain the value $i$ or something else?