Chapter 3: Key Takeaways
Summary Card
Core Idea
Probability theory is the mathematical language of prediction markets. Every market price is a probability, every trade is a disagreement about probability, and every profit or loss is the resolution of probabilistic uncertainty. Mastering the fundamentals --- from sample spaces to Bayesian updating --- gives you the tools to analyze any prediction market contract and make mathematically grounded decisions.
The Five Essential Concepts
| Concept | One-Sentence Summary | Key Formula |
|---|---|---|
| Conditional Probability | The probability of an event changes when you learn new information | P(A|B) = P(A and B) / P(B) |
| Bayes' Theorem | The exact recipe for updating beliefs with evidence | P(H|E) = P(E|H) P(H) / P(E) |
| Expected Value | The probability-weighted average outcome; your "edge" in a trade | EV = q - p |
| Variance | How spread out outcomes are; your "risk" in a trade | Var = p(1-p) for binary |
| Law of Large Numbers | Average outcomes converge to EV over many trials | Sample mean converges to population mean |
Key Formulas at a Glance
Complement: P(not A) = 1 - P(A)
Addition: P(A or B) = P(A) + P(B) - P(A and B)
Conditional: P(A|B) = P(A and B) / P(B)
Multiplication: P(A and B) = P(A|B) * P(B)
Bayes: P(H|E) = P(E|H) * P(H) / P(E)
Bayes (odds): posterior odds = likelihood ratio * prior odds
Independence: P(A and B) = P(A) * P(B) [iff independent]
Binary EV: EV = q - p (your prob minus market price)
Bernoulli Var: Var = p * (1 - p)
Beta update: Beta(a,b) + s successes, f failures -> Beta(a+s, b+f)
CLT std error: SE = sigma / sqrt(n)
Key Distributions for Prediction Markets
| Distribution | Use Case | Parameters |
|---|---|---|
| Bernoulli | Single binary contract outcome | p (probability) |
| Binomial | Number of wins in n independent trades | n (trials), p (win prob) |
| Beta | Uncertainty about a probability itself | alpha, beta (shape) |
| Normal | Profit distribution over many trades (via CLT) | mu (mean), sigma (std) |
Decision Checklist for Evaluating a Trade
- What is the sample space? What are all possible outcomes?
- What is my estimated probability q for this event?
- What is the market price p?
- Is there an edge? (q - p > 0 for a buy)
- How strong is my evidence? (Compute the likelihood ratio)
- What is the risk? (Variance, max loss, drawdown potential)
- How does this trade correlate with my other positions?
- Do I have enough bankroll to survive the variance?
- Am I making enough trades for the LLN to work in my favor?
Common Mistakes to Avoid
| Mistake | Why It Is Wrong | The Fix |
|---|---|---|
| Confusing P(A|B) with P(B|A) | Base rate fallacy / prosecutor's fallacy | Always apply full Bayes' theorem |
| Assuming independence | Correlated events amplify portfolio risk | Test for independence; model correlations |
| Ignoring variance | Positive EV does not mean guaranteed profit | Calculate risk alongside EV |
| Gambler's fallacy | Past losses do not make future wins more likely | Each independent trade is a fresh event |
| Assigning P = 0 or P = 1 | No evidence can update a certainty (Cromwell's rule) | Keep probabilities in [0.001, 0.999] |
| Overconfidence at extremes | Small calibration errors at 95%+ are very costly | Extra scrutiny for extreme probabilities |
Code Patterns to Remember
Quick Bayes update (odds form):
def quick_bayes(prior_prob, likelihood_ratio):
prior_odds = prior_prob / (1 - prior_prob)
posterior_odds = likelihood_ratio * prior_odds
return posterior_odds / (1 + posterior_odds)
Binary trade EV:
def trade_ev(market_price, your_prob, position_size=1.0):
return (your_prob - market_price) * position_size
Beta-Binomial update:
def beta_update(alpha, beta, successes, failures):
return alpha + successes, beta + failures
Bridge to Next Chapter
Chapter 3 gave you the mathematical theory of probability. Chapter 4 will give you the statistical practice --- how to estimate probabilities from data, test whether your edge is real, build confidence intervals, and assess whether prediction market prices are well-calibrated. The tools you built here (Bayesian updater, EV calculator, Monte Carlo simulator) will be extended and applied to real market data.