mathematics - The Minotaur's Labyrinth

You are trapped in a chamber in the center of the Minotaur's Labyrinth. There are $\mathbf{N}$ tunnels, $\mathbf{m}$ of which lead to safety; the remaining tunnels only lead back to the chamber. Each tunnel is of a different length, taking $h_i$ hours to travel. Each time you return to the chamber, the room shifts so that you can only choose tunnels at random.

1. What is the expected amount of time it will take you to escape?


2. You have 24 hours until the Minotaur wakes up. If there are 10 tunnels, such that $h_i = i$ (for $i$ = 1,2,...10) and two of the tunnels lead to safety do you believe you will escape in time?

Answer

1. You spend an expected $\frac{\sum h_i}N$ hours each time you travel a tunnel, and you have to travel an average of $\frac Nm$ tunnels to escape. The product is $\frac{\sum h_i}m$ hours total.


2. $\sum h_i=55$ and $m=2$, so the expected amount of time to escape is $\frac{55}{2}=27.5$ hours.