Expected waiting time at bus stop

We assume my arrival time is time is $12:00$ since I can set my arrival time anything with respect to reference.

Let the start of their working time be $\alpha$ minutes after 0:00 and $\beta$ minutes after 0:00 respectively.

$\alpha$ , $\beta$ leave remainders with 20, $x,y$ respectively.

Without loss of genrality let $x<y$ .

Our waiting time in this case is $x$

We want to find the average of this function for all possible values of $x,y$

Average= $\frac { \displaystyle \int _{ 0 }^{ 20 }{ \int _{ 0 }^{ y }{ xdxdy } } }{ \displaystyle \int _{ 0 }^{ 20 }{ \displaystyle \int _{ 0 }^{ y }{ dxdy } } }=\dfrac{20}{3} min=400 sec$

Assume the first bus arrives 0:00, and the second bus arrives at 0:x, where $x<20$ . You arrive from 0:00 to 0:20. Notice the wait time varies linearly, with a minimum at 0:10, where you wait an average of 5 minutes, and a max at the endpoints, with 10 minutes. Thus, the average can be seen as the weighted average, or $\frac {5*2+10}{3}$ minutes, or $\boxed{400}$ seconds