This is an extension of the puzzle The Minotaur's Labyrinth which is inspired by the comments from histocrat.
You are trapped (again) in a chamber at the center of the minotaur's labyrinth. There are $N$ tunnels, $m$ 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. However, this time you find the distribution of tunnel length inscribed on the wall. You still don't know the length of any particular tunnel, but you know the distribution of lengths.
You have 24 hours until the Minotaur wakes up. If you are already in a safe tunnel when he wakes up, you will escape safely.
Case 1 Suppose there are 10 tunnels such that $$h_i = \begin{cases} i, & i = 1, 2, \cdots 9 \\ 100, & i = 10 \end{cases}$$ and $2$ tunnels lead to safety.
- What is the optimal strategy you should use?
- Using the optimal strategy, what is the expected time it will take you to escape?
- Using the optimal strategy, what is the probability that you will escape?
Case 2 Suppose there are 10 tunnels such that $$h_i = 2^i$$ and $2$ tunnels lead to safety.
- What is the optimal strategy you should use?
- Using the optimal strategy, what is the expected time it will take you to escape?
- Using the optimal strategy, what is the probability that you will escape?
The strategy with the best probability of survival for case 2 will be accepted.
Using a computer (simulation) to answer questions 2 and 3 is allowed.
Some Clarifications
- A priori, each tunnel has an equal chance of being "safe"
- You only know that the tunnel you are in is "safe" once you have traveled to the end of it and realize that you have escaped the labyrinth.
- When you return to the chamber, the room shifts so that you have no knowledge even of which tunnel you just traveled.
- I admit, I'm unsure of how one would demonstrate "optimality" here. For now, I'm mostly interested in maximizing probability of survival. The current score to beat is 30.50% by ffao.