# Parrondo's Paradox

Two losing gambling games can be set up so that when they are played one after the other, they become winning. This paradox is named after the Spanish physicist Juan Parrondo who discovered how to construct such a scenario.

The simplest way is to use three biased coins. Imagine you are standing on stair zero, in the middle of a long staircase with 1001 stairs numbered from -500 to 500. You win if you can get to the top of the staircase, and the way you move depends on the outcome of flipping one of two coins. Heads you move up a stair, tails you move down a stair. In game 1, you use coin A, which is slightly biased and comes up heads 49.5% of the time and tails 50.5%. Obviously, these are losing odds. In game 2, you use two coins, B and C. Coin B comes up heads only 9.5% of the time, tails 90.5%. Coin C comes up heads 74.5% of the time, tails 25.5%. In game 2 if the number of the stair you are on at the time is a multiple of 3 (that is, ..., -9, -6, -3, 0, 3, 6, 9, 12, ...), then you flip coin B; otherwise you flip coin C. Game 2, it turns out, is also a losing game and would eventually take you to the bottom of the stairs. What Parrondo found, however, is that if you play these two games in succession in random order, keeping your place on the staircase as you switch between games, you will steadily rise to the top of the staircase!

## Reference

1. Harmer, G. P. and Abbott, D. "Losing Strategies Can Win by Parrondo's
Paradox." *Nature*, 402: 864 (1999).