The Parrondo Paradox
The Parrondo Paradox
Parrondo's paradox is a surprising statement about the combination of certain probabilistic events. We start with two games A and B that, individually, are losing games. But we define game C as follows: Flip a fair coin and play A if the coin comes up heads and B if it comes up tails. Then C can be a winning game.
The details: Game A is simply the flip of a coin that comes up heads with probability 0.495. Since heads win, the expected value of game A is clearly negative: a bettor who stakes $1 on each flip would have an expected long-term loss of 1¢ per game. Game B is a bit more complicated: Consider two coins, W and L, with W coming up heads with probability 0.745 and L coming up heads with probability 0.095. Clearly W is a winning coin and L is a losing coin. For game B, if the player's stake (it is assumed the player starts with $0 and wins or loses $1 on each throw) is divisible by 3, then coin L is used on the next flip; otherwise coin W is used. Some computation shows that the expected return from game B is negative: a player of this game will on average lose 0.87¢ per turn.
The surprise is that each of the following two games has a positive expected value.
The random game: Flip a fair coin and use the outcome to choose which of games A or B to play; the expected return is 1.57¢ per play.
BBABA: Play the games B, B, A, B, A in sequence; the expected return is a whopping 6.52¢ per play.
ABBB is a losing game but is unusual. If the number of plays is very large, the expected loss is about 0.27¢ per play. But for shorter games the game is a winning one; for plays lasting only 10 flips, the expected gain is 5.4¢ per play.