How many steps until the puzzle is solved?

Question

While solving a puzzle, a person is in one of two working states, Confused or On-track, until the puzzle is Solved. Each minute:

From Confused: stay Confused with probability $0.5$ , or become On-track with probability $0.5$ .
From On-track: slip back to Confused with probability $0.3$ , or reach Solved with probability $0.7$ .

Solved ends the process. The person starts Confused.

On average, how many minutes until the puzzle is Solved?

Accepted Answer

Let h(X) be the expected number of minutes to reach Solved, starting from state X. Once solved, no more time is needed, so we only need h(	ext{Confused}) and h(	ext{On-track}). The rule (first-step analysis for expected time): from any working state, one minute passes, then we add the expected time from wherever we land. h(	ext{Confused}) = 1 + 0.5\,h(	ext{Confused}) + 0.5\,h(	ext{On-track}), h(	ext{On-track}) = 1 + 0.3\,h(	ext{Confused}) + 0.7 \cdot 0. From the first equation, 0.5\,h(	ext{Confused}) = 1 + 0.5\,h(	ext{On-track}), so h(	ext{Confused}) = 2 + h(	ext{On-track}). Put th…

How many steps until the puzzle is solved?

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