Parrondo's Paradox Calculator
Explore how two losing games can combine to create a winning strategy
Simulate Parrondo's Paradox
Choose the game-playing strategy
Games played in each simulation run
Number of independent simulation runs
Initial capital amount
Game Rules
Game A (Losing Game)
• Simple coin flip with slightly unfavorable odds
• Probability of winning: 49.5%
• Win: +$1, Lose: -$1
• Expected value per game: -$0.01 (losing)
Game B (Also Losing Game)
• Two sub-games based on your current capital
Game B1 (when capital is multiple of 3):
→ Probability of winning: 9.5% (very unfavorable)
Game B2 (when capital is not multiple of 3):
→ Probability of winning: 74.5% (very favorable)
• Overall expected value: -$0.009 (losing)
The Paradox
• Combining these two losing games can create a winning strategy!
• Strategies like alternating (ABAB) or AABB patterns result in positive expected value
• The key is that mixing games allows you to spend more time in favorable states
• Even random selection between the two games produces a winning outcome
Strategy Guide
Game A Only
Always losing (-0.5% per game)
Game B Only
Always losing (-0.9% per game)
Alternating (AB)
Winning strategy! (~+1% per game)
AABB Pattern
Winning strategy! (~+0.5% per game)
Random Selection
Winning strategy! (~+0.8% per game)
Game Probabilities
Real-World Applications
Population dynamics in biology
Financial market strategies
Game theory research
Ratchet mechanisms in physics
Understanding Parrondo's Paradox
What is Parrondo's Paradox?
Parrondo's Paradox demonstrates the counterintuitive result that two losing games can combine to produce a winning game. This paradox was discovered by Spanish physicist Juan Parrondo while studying Brownian ratchets in physics.
Why Does It Work?
- •Game B has state-dependent probabilities based on your capital
- •Mixing strategies allows more time in favorable states
- •The correlation between games creates positive expected value
Mathematical Foundation
Markov Chain Analysis
The system can be modeled using states based on capital mod 3:
State 0: Capital ÷ 3 = 0 (play B1)
State 1: Capital ÷ 3 = 1 (play B2)
State 2: Capital ÷ 3 = 2 (play B2)
Steady State Probabilities
π₀ = 0.384 (unfavorable state)
π₁ = 0.154 (favorable state)
π₂ = 0.462 (favorable state)
Key Insights
Not Applicable to Casinos
Real casino games are designed to be independent, preventing this paradox from working
Requires Correlation
The games must be related through some external parameter (like current capital)
Broader Applications
Found in biology, economics, and physics where systems can switch between states