How long until the drone lands?

Question

A drone approaches a pad through three states, Far, Mid, and Close, until it is Landed. Each step:

From Far: always move to Mid.
From Mid: move to Close with probability $0.5$ , or drift back to Far with probability $0.5$ .
From Close: Land with probability $0.5$ , or drift back to Mid with probability $0.5$ .

Landing ends the process. The drone starts Far.

On average, how many steps until the drone Lands?

Accepted Answer

Let h(	ext{Far}), h(	ext{Mid}), h(	ext{Close}) be the expected steps to Land; landed needs 0 steps. First-step analysis: one step passes, then add the expected time from the next state. h(	ext{Far}) = 1 + h(	ext{Mid}), h(	ext{Mid}) = 1 + 0.5\,h(	ext{Close}) + 0.5\,h(	ext{Far}), h(	ext{Close}) = 1 + 0.5 \cdot 0 + 0.5\,h(	ext{Mid}). Substitute h(	ext{Far}) = 1 + h(	ext{Mid}) into the Mid equation: h(	ext{Mid}) = 1 + 0.5\,h(	ext{Close}) + 0.5\,(1 + h(	ext{Mid})) = 1.5 + 0.5\,h(	ext{Close}) + 0.5\,h(	ext{Mid}), so 0.5\,h(	ext{Mid}) = 1.5 + 0.5\,h(	ext{Close}), giving h(	ext{Mi…

How long until the drone lands?

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