The miner and the three tunnels

Question

A miner is trapped in a dark chamber with three tunnels. Tunnel A leads outside after $2$ hours of walking. Tunnel B loops back to the same chamber after $3$ hours. Tunnel C also loops back, after $5$ hours. Disoriented, each time he is in the chamber he picks one of the three tunnels uniformly at random (with equal probability $\tfrac13$ ), independently of past choices, he does not learn from his mistakes.

What is the expected total time until he escapes?

Accepted Answer

Let E be the expected time to escape, measured from the chamber. Condition on which tunnel he picks first: - Tunnel A (prob 	frac13): he is out after 2 hours. Contributes 2. - Tunnel B (prob 	frac13): he wanders 3 hours and lands back in the chamber, facing the same situation. Contributes 3 + E. - Tunnel C (prob 	frac13): he wanders 5 hours and returns. Contributes 5 + E. By the memoryless choice (returning to the chamber resets everything), E = 	frac13(2) + 	frac13(3 + E) + 	frac13(5 + E). Multiply out: E = \frac{2 + 3 + 5}{3} + \frac{2}{3}E = \frac{10}{3} + \frac{2}{3}E. Then E - 	fr…

The miner and the three tunnels

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