Coach or Kasparov

To encourage John’s promising chess career, his coach offers him a prize if he wins (at least) two games in a row in a three-game series to be played with his coach and Garry Kasparov alternately: coach-kasparov-coach or kasparov-coach-kasparov, according to John’s choice. Kasparov is (evidently) a better player than John’s coach. Which series should John choose? ¹

Let $p_C$ be the probability of winning the coach and $p_K$ the probability of winning Kasparov, where $p_C < p_K$ . The winning outcomes for the first scenario (C-K-C) are:

outcome	probability
$C_0K_1C_1$	$p_C~p_K~(1-p_C)$
$C_1K_1C_0$	$p_C~p_K~(1-p_C)$
$C_1K_1C_1$	$p_C~p_K~p_C$

And thus $p_C~p_K~(1-p_C) + p_C~p_K~(1-p_C) + p_C~p_K~p_C = p_K~p_C~(2 - p_C)$. For the second scenario (K-C-K):

outcome	probability
$K_0C_1K_1$	$p_K~p_C~(1-p_K)$
$K_1C_1K_0$	$p_K~p_C~(1-p_K)$
$K_1C_1K_1$	$p_K~p_C~p_K$

Where $p_K~p_C~(1-p_K) + p_K~p_C~(1-p_K) + p_K~p_K~p_C = p_K~p_C~(2 - p_K)$. Hence:

$\begin{align*} p_K~p_C~(2 - p_K) & \stackrel{?}{=} p_K~p_C~(2 - p_K) \\ 2 - p_C & \stackrel{?}{=} 2 - p_K \\ p_C & < p_K \end{align*}$

This is adapted from problem 2 of Frederick Mosteller’s “Fifty Challenging Problems in Probability”. ↩︎