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 threegame series to be played with his coach and Garry Kasparov alternately: coachkasparovcoach or kasparovcoachkasparov, according to John’s choice. Kasparov is (evidently) a better player than John’s coach. Which series should John choose? ^{1}
Let be the probability of winning the coach and the probability of winning Kasparov, where . The winning outcomes for the first scenario (CKC) are:
outcome  probability 

$C_0K_1C_1$  $p_C~p_K~(1p_C)$ 
$C_1K_1C_0$  $p_C~p_K~(1p_C)$ 
$C_1K_1C_1$  $p_C~p_K~p_C$ 
And thus $p_C~p_K~(1p_C) + p_C~p_K~(1p_C) + p_C~p_K~p_C = p_K~p_C~(2  p_C)$. For the second scenario (KCK):
outcome  probability 

$K_0C_1K_1$  $p_K~p_C~(1p_K)$ 
$K_1C_1K_0$  $p_K~p_C~(1p_K)$ 
$K_1C_1K_1$  $p_K~p_C~p_K$ 
Where $p_K~p_C~(1p_K) + p_K~p_C~(1p_K) + p_K~p_K~p_C = p_K~p_C~(2  p_K)$. Hence:

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