Will the robot reach the goal or the pit?

Question

A robot moves among rooms. From the Start it goes to room A or room B, each with probability 0.5. From A it reaches the Goal with probability 0.5 or returns to Start with probability 0.5. From B it falls in the Pit with probability 0.6 or returns to Start with probability 0.4. Goal and Pit end the walk. Starting from Start, what is the probability the robot reaches the Goal before the Pit?

Accepted Answer

Let f(X) be the probability of reaching Goal before Pit, starting from room X. The two ending rooms are known: f(	ext{Goal}) = 1 and f(	ext{Pit}) = 0. For every other room, we use first-step analysis: the chance of eventually winning is the average of the winning chances of wherever the next step lands. f(A) = 0.5 \cdot 1 + 0.5 \cdot f(	ext{Start}) = 0.5 + 0.5\,f(	ext{Start}), f(B) = 0.6 \cdot 0 + 0.4 \cdot f(	ext{Start}) = 0.4\,f(	ext{Start}), f(	ext{Start}) = 0.5\,f(A) + 0.5\,f(B). Substitute the first two lines into the third: f(	ext{Start}) = 0.5\,(0.5 + 0.5\,f(	ext{Start})) + 0.…

Will the robot reach the goal or the pit?

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