Lost in a cave
You are in a dark cave with three exits that look the same. Each time you are in the cave, you pick one of the three uniformly at random, without remembering earlier choices.
- One exit leads outside after 3 hours of walking.
- One leads back into the cave after 1 hour.
- One leads back into the cave after 2 hours.
Find the expected time, in hours, until you are outside.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Condition on the first exit you choose. After coming back, the situation is the same as at the start.
Let be the answer. The second and third exits cost 1 and 2 hours, then more.
h
s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Every return to the cave restarts the problem, so the expected time appears on both sides of a first-step equation.
Step 1.
Step 2. , so hours.
Check. On average you need 3 tries to find the right exit, so 2 wrong ones, each costing hours on average, plus the 3 hours of the right exit: .