The frog, the rock, and the heron

Question

Five lily pads sit in a row, numbered 1 through 5. Pad 1 touches a safe rock; pad 5 has a hungry heron. A frog starts on pad 2. Each minute it hops to an adjacent pad, left or right with probability 	frac12 each. It stops when it reaches the rock (pad 1, safe) or the heron (pad 5, eaten). What is the probability the frog reaches safety, and how many hops does it take on average before the game ends?

Accepted Answer

This is a symmetric gambler's ruin on pads 1,\dots,5, with absorbing barriers at both ends. Label safety as the barrier at position 1 and doom at position 5. Survival probability. For a fair walk between barriers at a and b, the probability of hitting a before b from position x is the linear interpolation \frac{b - x}{b - a}. With a = 1 (safe), b = 5 (heron), starting at x = 2: P(	ext{reach pad }1	ext{ first}) = \frac{5 - 2}{5 - 1} = \frac{3}{4}. So the frog survives with probability 	frac34. Makes sense, it starts right next to the rock. Expected number of hops. For a symmetric walk, the …

The frog, the rock, and the heron

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