A frog hopping between three lily pads

Question

Three lily pads sit at the corners of a triangle, each pad connected to the other two. A frog starts on one pad; each hop it jumps to one of the other two pads, each with probability $\tfrac12$ . It stops when it reaches a particular target pad.

What is the expected number of hops to reach the target pad, starting from another pad?

Accepted Answer

By symmetry, the two non-target pads have the same expected hop count e. From a non-target pad, one hop reaches the target (value 0) with probability 	frac12, or the other non-target pad (value e) with probability 	frac12: e = 1 + 	frac12 (0) + 	frac12 e \;\Longrightarrow\; 	frac12 e = 1 \;\Longrightarrow\; e = 2. Takeaway. On a triangle every non-target pad looks the same, so one equation suffices: the target is reached in 2 hops on average.

A frog hopping between three lily pads

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