  Penney's game is a two player zero-sum game in which each player chooses a
three-flip pattern of heads and tails and the winner is the player whose
pattern occurs first in repeated tosses of a fair coin. Because the players
choose sequentially, the second mover has the advantage. In fact, for any
three-flip pattern, there is another three-flip pattern that is strictly more
likely to occur first. This paper provides a novel no-arbitrage argument that
generates the winning odds corresponding to any pair of distinct patterns. The
resulting odds formula is equivalent to that generated by Conway's "leading
number" algorithm. The accompanying betting odds intuition adds insight into
why Conway's algorithm works. The proof is simple and easy to generalize to
games involving more than two outcomes, unequal probabilities, and competing
patterns of various length. Additional results on the expected duration of
Penney's game are presented. Code implementing and cross-validating the
algorithms is included.
