  Toral (2002) considered an ensemble of N\geq2 players. In game B a player is
randomly selected to play Parrondo's original capital-dependent game. In game
A' two players are randomly selected without replacement, and the first
transfers one unit of capital to the second. Game A' is fair (with respect to
total capital), game B is losing (or fair), and the random mixture
{\gamma}A'+(1-{\gamma})B is winning, as was demonstrated by Toral for
{\gamma}=1/2 using computer simulation. We prove this, establishing a strong
law of large numbers and a central limit theorem for the sequence of profits of
the ensemble of players for each {\gamma}\in(0,1). We do the same for the
nonrandom pattern of games (A')^r B^s for all integers r,s\geq1. An unexpected
relationship between the random-mixture case and the nonrandom-pattern case
occurs in the limit as N\rightarrow\infty.
