2 if the previous pull was lost, and with probability p (< 1 2 ) if won. Show that the probability un that the player wins at the nth pull satisfies un + ( 1 2 − p)un−1 = 1 2 for n > 1.

Deduce that un =

1 + (−1)n−1( 1 2 − p)n 3 − 2p for n ≥ 1.