Question
Just like in anime, you are taking a train in Tokyo. There are two train lines, say $A$ and $B$. The time (in minutes) between consecutive trains arriving on line $A$ is $\exponential{\lambda_1}$ distributed. Similarly, the time between consecutive trains arriving on line $B$ is $\exponential{\lambda_2}$ distributed, independent of line $A$. You arrive at a uniformly random time throughout the day and board the next train that arrives, regardless of the line. Find the expected amount of time you wait (after arriving at the station) until the next train arrives. Report the answer when $\lambda_1 = 1/10$ and $\lambda_2 = 3/20$.