Lecture Notes: Probability, Expected Value, and Betting Strategies

Game Scenarios

Game 1: Negative Expected Value

• Probability of winning: 1/3
• Winning payout: 1.8 times the wager
• Losing payout: lose entire wager
• Expected value calculation results in a negative value → no long-term winning strategy.
• Conclusion: Continuous playing leads to losing all money.

Game 2: Guaranteed Winning

• 100% chance of winning; payout is a double of the wager.
• This scenario acts as a "money printer".
• Winning strategy: Bet entire savings or use margin trading.
• Conclusion: This scenario allows for guaranteed profit.

The Real Interesting Problem

• Situation: Positive expected payout but no guarantee of winning.
• Example: Investing in S&P500 Index Fund
  • Probability of win based on historical data: 59%.
  • Strategy: Sell at 100% profit (price doubles) and cut losses at 25% drop.
  • Expected payout per play is positive.
• Optimal risk and strategies discussion:
  • Risk everything (strategy from WallStreetBets).

Defining the Problem

• Game parameters:
  • p = probability of winning
  • q = probability of losing = 1 - p
  • Risk a fixed percentage r of the portfolio.
  • Gain tr for each win, lose sr for each loss.
  • Auxiliary conditions: can’t lose more than entire portfolio, not guaranteed to win.

Kelly Criterion

• Optimal risk: r = (p/t) - (q/s)
  • Proportional relationship: Increased winning probability (p) leads to higher risk.
  • Inversely proportional: Increased loss probability (q) or loss amount (s) leads to lower risk.

Different Types of Averages

• Arithmetic Mean: Total values / number of values
• Quadratic Mean (Root Mean Square):
  • Applying squares and square roots; useful in certain contexts.
• Harmonic Mean (for rates):
  • Like weights or cycles, useful in job completion rates.
• Geometric Mean: Used for measuring compounded growth; reflects multiplicative processes.

Applications of Different Averages

• Wealth Distribution: Arithmetic mean can be skewed by outliers (e.g., billionaires).
  • Median is a better representative avoiding outlier distortion.
• Compound Growth: Use geometric mean to calculate average growth; reflects the true increase.

Expected Value of Random Variables

• Definition: The central tendency of outcomes is represented mathematically.
  • Sum of (value of random variable) * (probability of that value).
• Binomial Distribution Example: X = number of wins in trials (with success probability p).
• Expected Value of Binomial: Generally found through simulations and back-testing.*

Transformations of Random Variables

• Y as a transformation can reflect characteristics of X.
• Different transformations can yield new results and expected values; key in deriving complex outcomes.

Maximizing Expected Value Strategy

• Challenging Examples: Often maximization strategies don't yield sustainable results (e.g., betting all on a coin flip).
• Median & Mode as Alternatives: May provide better long-term strategies over merely maximizing expected outcomes.

Conclusion

• Kelly Criterion suggests optimal percentages to keep risk manageable while seeking profit.
• Balance and strategy adjustment based on probabilistic expectations yield more reliable outcomes.
• Understanding varied means contributes to a more informed decision-making process in investing.