# The Parrondo Paradox

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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.

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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.

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Contributed by: Stan Wagon (July 2011)
(Macalester College)
Open content licensed under CC BY-NC-SA

## Details

In the Demonstration, the lengths of the games are 500 flips and the results are averaged over 100 repetitions. Snapshots 1 and 2 show that games A and B are losing games. Snapshot 3 shows that the combination ABBAB is a winning game.

For more information on the paradox, which arose from some considerations of physics, see Parrondo's paradox or the MathWorld Parrondo's paradox (Wolfram MathWorld) entry. For detailed analysis of the games discussed here, see [1].

Reference

[1] D. Velleman and S. Wagon, "Parrondo's Paradox," Mathematica in Education and Research, 9(3–4), 2001 pp. 85–90.

## Permanent Citation

Stan Wagon

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