Lecture on the Martingale Strategy

Introduction

• Topic: Gambling strategy in a fictional casino with roulette (50/50 chance on red or black)
• Strategy: Martingale Strategy
• Concept: Double down on every loss, aiming for eventual win

How the Martingale Strategy Works

• Betting Process:
  • Step 1: Bet 1 pound, lose -> proceed to next step
  • Step 2: Bet 2 pounds, lose -> proceed to next step
  • Step 3: Bet 4 pounds, lose -> continue doubling the last bet
  • Continue doubling until a win
• Win Condition: Bet on black, double the money if win, otherwise follow the doubling strategy

Mathematical Foundations

• Bet Calculation:
  • Bets increase by powers of 2
  • Formula: 2^(k-1)
• Total Loss Calculation (Geometric Series):
  • Sum of losses: 1 + 2 + 4 + ... + 2^(k-1)
  • Geometric series sum formula: Sum = 2^k - 1
• Next Bet (Post Loss): 2^k
• Profit: If win after a series of losses, profit = 1 pound

Practical Issues with Martingale Strategy

• Non 50/50 Games: e.g., Zero/Double zero in roulette alters odds
• Maximum Bet Limits in Casinos: Prevents continued doubling
• Exponential Growth: Bets grow quickly, reaching limits fast
• Reality Check: Human error in assessing streak loss probability

Probability Calculations

• Rich vs Poor: More money reduces the probability of losing all funds
• Probability of k Losses: 1 / 2^k (1 over 2 to the power k)
• General Formula: 1 - (1 / n) for probability of winning 1 pound
• Examples:
  • n = 100 pounds: 99% chance to win 1 pound, 1% to lose 100 pounds
  • To double money: Calculate win probability for n times -> (1 - 1/n)^n
  • Limits: 1/e (36.8%) regardless of the initial money

Conclusion

• Final Insight: Straight bet on black gives 50% chance to double money compared to Martingale's 36.8%
• Reality of Gambling Strategies: All strategies break down due to human, financial, or casino constraints.

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