Late lecturer making a mile in random dashes

A lecturer must cover $1$ mile in $10$ minutes. He makes up to $3$ dashes. On dash $n$ he covers $x_n$ miles, uniform on $[0,1]$, taking $t_n$ minutes, uniform on $[0,10]$, with all these values independent. After each dash he starts another only if he has not yet covered a mile and there is still time left, with at most $3$ dashes in total. If the $10$ minutes expire part-way through a dash, time freezes and he still completes that dash's full distance. He is on time if his total distance reaches at least $1$ mile. What is the probability he is on time?

$2/3$. Measure time in units of $10$ minutes. The first dash never uses up all the time, so a second dash always happens, and he arrives if $X_1+X_2\ge1$ (probability $\tfrac12$). Otherwise he gets a third dash only if $T_1+T_2<1$ (probability $\tfrac12$), and then needs $X_1+X_2+X_3\ge1$, which given $X_1+X_2<1$ happens with probability $\tfrac23$. Total: $\tfrac12+\tfrac12\cdot\tfrac12\cdot\tfrac23=\tfrac23$.