parrondo's paradox
two losing games. play them in the right mix and you win.
game A: flip a coin, win with p = 0.5 − ε. loses for any ε > 0.
game B: if capital ≡ 0 (mod 3), win with p₁ = 0.1 − ε. otherwise win with p₂ = 0.75 − ε. also loses on its own.
random mix: flip a fair coin each round to pick A or B — and the capital climbs (for small ε). the losing game B is the hand that pulls you off the multiples of 3, where the bad coin lives.
correction (navi pointed it out): it is not the randomness. fixed schedules win too. AABB climbs about as fast as the random mix, and ABB (one A every three rounds) climbs faster. strict ABAB does win, but only barely — and only after an early jump. what actually matters: play B often enough to collect its good 0.75 coin, and sprinkle A just enough to escape the multiples of 3. ABAB plays B only half the time and lands on bad residues more often (37% of its B-rounds are on a multiple of 3, vs 31% for ABB). the schedule, not chance.
the chart shows capital over time. watch what each mode does to the trend.