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+# Betting Theory Fundamentals
+
+This resource explains the core theoretical foundations of rational betting, expected value, variance management, and market efficiency.
+
+**Foundation for:** All betting and forecasting decisions
+
+---
+
+## Why Learn Betting Theory
+
+**Core insight:** Betting theory separates decision quality from outcome quality. Make +EV decisions repeatedly and survive variance.
+
+**Enables:**
+- Think probabilistically (convert beliefs to quantifiable edges)
+- Manage risk rationally (distinguish bad decisions from bad outcomes)
+- Avoid costly mistakes (identify predictable failure modes)
+- Optimize long-term growth (balance aggression with preservation)
+
+**Research foundation:** Kelly (1956), Samuelson (1963), Thorp (1969), behavioral economics (Kahneman & Tversky), market efficiency (Fama).
+
+---
+
+## 1. Expected Value Framework
+
+### Definition and Formula
+
+**Expected Value (EV):** Probability-weighted average of all possible outcomes.
+
+```
+EV = Σ(Probability × Outcome)
+
+Binary bet:
+EV = (P_win × Amount_won) - (P_lose × Amount_lost)
+```
+
+**Example:**
+```
+Bet $100 on 60% event at even odds (+100)
+EV = (0.60 × $100) - (0.40 × $100) = $20
+EV% = +20% per $100 wagered
+```
+
+### Positive vs Negative EV
+
+**Decision Framework:**
+- **EV > +5%:** Strong bet (after fees/uncertainty)
+- **EV = 0% to +5%:** Marginal (consider passing)
+- **EV < 0%:** Never bet (unless hedging)
+
+**Critical Rule:** Judge decisions by EV, not outcomes. Good decisions lose sometimes; bad decisions win sometimes. Process matters in small samples, results matter over 100+ trials.
+
+### Converting Market Odds to EV
+
+**Step 1: Implied probability**
+```
+Decimal odds: P = 1 / Odds
+  Example: 1.67 → 60%
+
+American (+): P = 100 / (Odds + 100)
+  Example: +150 → 40%
+
+American (-): P = |Odds| / (|Odds| + 100)
+  Example: -150 → 60%
+```
+
+**Step 2: Calculate edge**
+```
+Your probability: 70%
+Market probability: 60%
+Edge = 70% - 60% = +10%
+```
+
+**Step 3: Calculate EV**
+```
+Bet $100 at 1.67 odds:
+EV = (0.70 × $67) - (0.30 × $100) = +$16.90 = +16.9%
+```
+
+### Law of Large Numbers
+
+**Key principle:** Observed frequency converges to true probability as sample size increases.
+
+**Practical thresholds:**
+- 10 bets: High variance, might be down despite +EV
+- 100 bets: Convergence starting, likely near EV
+- 1000 bets: Results tightly centered around EV
+
+**Application:** Don't judge strategy on <30 trials. Variance dominates small samples.
+
+---
+
+## 2. Variance and Risk
+
+### Standard Deviation
+
+**Measures outcome dispersion around EV.**
+
+**Formula:**
+```
+σ = √(P_win×(Win-EV)² + P_lose×(Loss-EV)²)
+```
+
+**Example ($100 bet, 60% win, even odds):**
+```
+EV = $20
+σ = √(0.60×(100-20)² + 0.40×(-100-20)²)
+σ = √9600 = $98
+
+Coefficient of Variation: σ/EV = $98/$20 = 4.9
+```
+
+**Interpretation:** Standard deviation ($98) is 5× the EV ($20). Variance dominates signal.
+
+### Volatility Categories
+
+**Coefficient of Variation (CV = σ/EV):**
+- CV < 1: Low volatility (10-30 trials to see EV)
+- CV = 1-3: Moderate (30-50 trials)
+- CV = 3-10: High (50-100 trials)
+- CV > 10: Extreme (100+ trials)
+
+**Higher CV requires:** Larger bankroll, more patience, stronger discipline.
+
+### Risk of Ruin
+
+**Probability of losing entire bankroll before profit.**
+
+**Practical Guidelines:**
+
+| Bet Size | Risk of Ruin | Assessment |
+|----------|--------------|------------|
+| 50% of bankroll | ~40% | Reckless |
+| 25% of bankroll | ~20% | Aggressive |
+| 10% of bankroll | ~5% | Moderate |
+| 5% of bankroll | ~1% | Conservative |
+| 2% of bankroll | ~0.1% | Very conservative |
+
+**Kelly Criterion naturally manages risk of ruin. Never bet >10% of bankroll on single bet.**
+
+### Managing Volatility
+
+**1. Fractional Kelly (Primary Tool):**
+- Full Kelly: 100% variance, 40%+ drawdowns
+- Half Kelly: 25% variance, ~20% drawdowns
+- Quarter Kelly: 6% variance, ~10% drawdowns
+
+**2. Diversification:**
+- Multiple uncorrelated +EV bets
+- Requires independence (correlation < 0.3)
+
+**3. Expected Drawdown:**
+- Even optimal betting experiences 20-40% drawdowns
+- Mentally prepare for temporary losses
+- Don't confuse drawdown with -EV strategy
+
+---
+
+## 3. Bankroll Management
+
+### Defining Your Bankroll
+
+**Valid:** Money you can afford to lose entirely, separate from emergency fund, investment portfolio, daily expenses. **Starting:** $500-$5000 recreational, $10,000+ serious.
+
+**NOT valid:** Money needed for bills, emergency fund, retirement, money you'd be devastated to lose.
+
+### Separation Principle
+
+**Why:** Prevents scared money and revenge betting. Clear accounting, tax clarity, risk containment.
+
+**Implementation:** Separate betting account, never add money mid-downswing, withdraw profits periodically, stop if bankroll → $0.
+
+### Growth vs Preservation
+
+**Preservation (Default):** 1/4 to 1/2 Kelly, for most bettors and bankrolls <$5000
+**Growth (Advanced):** 1/2 to full Kelly, for large bankrolls and high variance tolerance (requires 2+ years track record)
+
+### Dynamic Sizing
+
+Bet size scales with bankroll. Example: $1000 bankroll at 5% = $50. After wins → $1500 → bet $75. After losses → $600 → bet $30.
+
+**Recalculate:** Daily if >20% change, weekly (active), monthly (casual).
+
+### Withdrawal Strategy
+
+**Recommended:** When bankroll doubles, withdraw original amount, continue with profit (break-even if lose profit).
+**Conservative:** 50% profit monthly. **Aggressive:** Never withdraw (full compounding).
+
+---
+
+## 4. Market Efficiency
+
+### Efficient Market Hypothesis
+
+**Core claim:** Prices reflect all available information. **Reality:** Semi-strong efficient in liquid, mature markets.
+
+**Market knows:** Published polls/news, historical base rates, expert commentary, obvious statistical patterns.
+
+### Where Edges Exist
+
+**1. Information Asymmetry:** Local knowledge, domain expertise
+**2. Model Superiority:** Better statistical model, proper extremizing
+**3. Lower Transaction Costs:** Market 5% fee vs your 0-1%
+**4. Behavioral Biases:** Recency bias, base rate neglect, narrative following
+**5. Market Immaturity:** Low liquidity, niche topics, few informed traders
+
+**Before betting, ask:** "What information or model do I have that the market doesn't?"
+- Nothing → Pass | Vague → Pass | Specific → Investigate
+
+### Trust vs Question Market
+
+**Trust:** Liquid, mature, objective outcome, many informed participants, low emotion
+**Question:** Illiquid, new, subjective outcome, few informed participants, high emotion (politics, fandom)
+
+---
+
+## 5. Common Betting Mistakes
+
+### Chasing Losses
+**What:** Increasing bet size after losses. **Why:** Loss aversion, emotional arousal.
+**Fix:** Never increase bet size after loss, use bankroll %, take break after 2+ losses.
+
+### Tilt (Emotional Betting)
+**Triggers:** Bad beat, streaks, external stress. **Symptoms:** No analysis, ignoring Kelly, revenge betting.
+**Fix:** Pre-commit no bets when tilted. Checklist: Calm? Calculate EV? Kelly sizing? Betting for +EV not revenge?
+
+### Overconfidence Bias
+**What:** Overestimating probability accuracy (90% when true is 70%).
+**Fix:** Track calibration, log predictions + outcomes, calculate curve quarterly. Do 70% predictions happen 70%?
+
+### Ignoring Variance
+**What:** Judging strategy on <30 trials. Example: "Down 15% after 20 bets, strategy sucks" (normal variance).
+**Fix:** Require 50+ bets minimum, 100+ preferred, 200+ for high confidence.
+
+### Outcome Bias
+**What:** Judging by results not process. +15% EV lost = good decision (bad outcome). -10% EV won = bad decision (lucky).
+**Fix:** Checklist: EV correct? Edge > threshold? Kelly fraction? Followed system? YES = good decision regardless of outcome.
+
+### Hindsight Bias
+**What:** After outcome, "I knew it would happen."
+**Fix:** Pre-commit logging, write probability before event, don't revise after, accept 40% events happen 40%.
+
+---
+
+## 6. Integration with Kelly Criterion
+
+### EV Drives Kelly
+
+**Kelly derives from:** Expected value (edge), odds received, bankroll optimization (maximize log wealth).
+
+**Key relationship:** `f* = (bp - q) / b`. Edge drives bet size: 10% edge → ~10% Kelly, 5% edge → ~5% Kelly, 0% edge → 0% bet.
+
+### Variance Tolerance
+
+| Fraction | Variance | Growth | Drawdown |
+|----------|----------|--------|----------|
+| Full (1.0) | 100% | 100% | ~40% |
+| Half (0.5) | 25% | 75% | ~20% |
+| Quarter (0.25) | 6% | 50% | ~10% |
+
+### Bankruptcy Protection
+
+Kelly never bets 100%: prevents ruin, keeps capital for next bet, scales down as bankroll shrinks. **Practical:** Stop if bankroll drops 80-90%.
+
+---
+
+## 7. Practical Examples for Forecasters
+
+### Example 1: Election Prediction Market
+
+**Scenario:** Market 55%, your forecast 65%, bankroll $2000
+
+**Step 1: Edge**
+```
+Edge = 65% - 55% = +10%
+Threshold: 5%
+Decision: +10% > 5% → Proceed
+```
+
+**Step 2: EV**
+```
+Bet $100 at 1.82 odds → Win $82
+EV = (0.65 × $82) - (0.35 × $100) = +$18.30 = +18.3%
+```
+
+**Step 3: Kelly**
+```
+Full Kelly: 22.3%
+Half Kelly: 11.2%
+Bet: $2000 × 11.2% = $224
+```
+
+### Example 2: Brier Score Tracking
+
+**50 forecasts, goal: Brier < 0.15**
+
+| Forecast | Your P | Outcome | (P-O)² |
+|----------|--------|---------|--------|
+| Event A | 80% | YES (1) | 0.04 |
+| Event B | 30% | NO (0) | 0.09 |
+| Event C | 90% | YES (1) | 0.01 |
+| Event D | 60% | NO (0) | 0.36 |
+| Event E | 70% | YES (1) | 0.09 |
+
+**Brier:** 0.59 / 5 = 0.118 (Excellent)
+
+**Analysis:** Event D large error normal (40% events happen). Don't game metric by avoiding 60% predictions.
+
+### Example 3: Extremizing
+
+**Forecasts:** You 72%, A 68%, B 75%, C 70%, Market 71%
+**Average:** 71.2%
+
+**Extremize:**
+```
+Factor: 1.3 (moderate)
+Extremized = 50% + (71.2% - 50%) × 1.3 = 77.6% ≈ 78%
+
+Edge: 78% - 71% = +7%
+Half Kelly ≈ 3.5% of $5000 = $175 bet
+```
+
+### Example 4: Correlated Portfolio
+
+**Scenario:** Democrats House (60% yours, 55% market) + Senate (55% yours, 50% market)
+**Correlation:** 0.7 (high)
+
+**Naive (WRONG):**
+```
+Bet A: 5% × $10k = $500
+Bet B: 5% × $10k = $500
+Total: $1000 (10%)
+```
+
+**Correct:**
+```
+Adjust for correlation: 1 - (0.7 × 0.5) = 0.65
+Bet A: $500 × 0.65 = $325
+Bet B: $500 × 0.65 = $325
+Total: $650 (6.5%)
+```
+
+**Reasoning:** Positive correlation amplifies risk. Reduce sizing to maintain tolerance.
+
+---
+
+## Key Takeaways
+
+### The 10 Commandments
+
+1. **Expected Value is King** - Judge decisions by EV, not outcomes
+2. **Variance is Inevitable** - Embrace it; don't fight it
+3. **Bankroll is Sacred** - Protect it above all else
+4. **Kelly is Your Guide** - But use fractional (1/4 to 1/2)
+5. **Market is Usually Right** - You need edge to beat it
+6. **Discipline Over Impulse** - System beats emotion
+7. **Sample Size Matters** - 50+ bets before judgment
+8. **Calibration is Honesty** - Track it religiously
+9. **Correlations Kill** - Adjust for portfolio risk
+10. **Survival Enables Profit** - Can't win if bankrupt
+
+### Mental Models
+
+**Betting = Business**
+- Bankroll = Working capital
+- EV = Profit margin
+- Variance = Market volatility
+- Kelly = Capital allocation
+
+**Decision Quality ≠ Outcome Quality**
+- Good decisions lose sometimes (variance)
+- Bad decisions win sometimes (luck)
+- Process > Results (small samples)
+- Results > Process (large samples 100+)
+
+### Integration Workflow
+
+**Before betting:**
+1. Make forecast (Bayesian, reference class)
+2. Calculate edge vs market
+3. Check edge > threshold (5%+)
+4. Use Kelly for sizing
+5. Execute and log
+
+**After betting:**
+1. Track outcome
+2. Update calibration
+3. Calculate Brier score
+4. Don't judge single bet
+5. Evaluate after 50+ bets
+
+---
+
+**Return to:** [Main Skill](../SKILL.md#interactive-menu)