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.
- What is the expected amount of time it will take you to escape?
- 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
- 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.
- $\sum h_i=55$ and $m=2$, so the expected amount of time to escape is $\frac{55}{2}=27.5$ hours.
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