It has been shown that it is possible to construct two games that when played individually lose, but alternating randomly or deterministically between them can win. This apparent paradox has been dubbed “Parrondo’s paradox.” The original games are capital-dependent, which means that the winning and losing probabilities depend on how much capital the player currently has. Recently, new games have been devised, that are not capital-dependent, but historydependent. We present some analytical results using discrete-time Markovchain theory, which is accompanied by computer simulations of the games. ; http://trove.nla.gov.au/work/16428787 ; Papers based on presentations at the 9th International Symposium on Dynamic Games and Applications held in Adelaide, South Australia in Dec. 2000