3.6 Flips, Not 4, and the Beautiful Argument That Says Otherwise
A pebble climbing four boxes on coin flips needs 18/5 flips on average, and the two-line renewal argument that gives 4 is wrong. Its premise is true, since half of all games really do end on flip two, but the non-finishing half is two different states: tails-tails sends the pebble home while heads-heads leaves it on box 3, one flip from the exit and worth only 14/5.