Probability & Statistics

What you will learn

Expectation Formula

E = (Win% × Avg Win) - (Loss% × Avg Loss). This is THE most important metric. If E > 0, your strategy is profitable long-term. Example: 50% win rate, avg win $200, avg loss $100 → E = (0.5 × 200) - (0.5 × 100) = $50 per trade.

Why It's The Key Metric

You can have 30% accuracy and be profitable if your wins are 3x your losses. Or have 70% accuracy and lose money if your losses are larger than wins. Expectation summarizes everything.

Concept

Your edge only manifests with enough trades. 10 trades aren't enough. 100 trades start being representative. 300+ trades give statistical confidence. Short-term, luck dominates. Long-term, probability converges.

Short-Term Variance

Even with 60% win rate, you can have 5-7 consecutive losses. This does NOT mean your strategy doesn't work. It's normal variance. That's why capital management matters: to survive bad streaks.

Reality vs Theory

Books assume normal distribution (bell curve). Trading reality has "fat tails": extreme events happen more often than expected. Prepare for black swans (unexpected crashes).

Outliers Impact

Sometimes 80% of your annual profits come from 20% of your trades (Pareto Principle). You can't predict which ones. That's why NEVER skip a trade from your strategy: you might miss the winning outlier.

Losing Streak Probability

Formula: P(N losses) = (Loss Rate)^N. Example with 40% loss rate: 1 loss = 40%, 2 consecutive = 16%, 3 consecutive = 6.4%, 5 consecutive = 1%. With 100 trades, you'll statistically have at least one 5-loss streak.

Psychological Impact

Knowing that 5-7 loss streaks are statistically normal helps you NOT abandon your strategy prematurely. Average trader quits their system right before the winning streak arrives.

Volatility of Your Results

Standard deviation measures how much your results vary. Low deviation = consistent results. High deviation = erratic results. Two traders with same expectation can have VERY different experiences based on their standard deviation.

Sharpe Ratio

Formula: (Return - Risk-free rate) / Standard Deviation. Measures risk-adjusted return. Sharpe > 1 = good. Sharpe > 2 = excellent. Sharpe > 3 = exceptional. Prefer high Sharpe strategies: less stress, more consistency.

Correlation Coefficient

Measures relationship between two assets. +1 = move together. 0 = independent. -1 = move opposite. BTC and ETH have ~0.8-0.9 correlation (high). BTC and USD have negative correlation (~-0.3).

Smart Diversification

Trading 5 assets isn't enough. If all have +0.9 correlation, it's like trading 1 asset. Seek assets with <0.3 correlation for true diversification. Example: Crypto + Bonds + Gold have low correlation.

Win Rate

Winning trades / Total trades. 60% win rate is excellent. 50% is good. <40% requires very high R:R. CAUTION: Win rate alone means nothing. A 90% win rate system can go bankrupt if the 10% loss is huge.

Profit Factor

Gross profits / Gross losses. PF > 1.5 = good. PF > 2 = excellent. PF > 3 = exceptional. Example: $10,000 won, $5,000 lost → PF = 2. Means you win $2 for every $1 you lose.

Average R-Multiple

1R = your initial risk (distance to stop loss). If you risk $100 and win $300, it's 3R. If you lose, it's -1R. Average R-multiple > 0.5 is good. > 1 is excellent. Allows comparing strategies across markets.

Maximum Drawdown

Maximum drop from peak to valley. If your account went from $10,000 to $7,000, MDD = 30%. Drawdowns of 20-30% are normal even for good traders. >50% is dangerous. Plan for worst-case scenario.

Required Sample Size

Minimum 100 trades to start having confidence. 300+ trades for statistical confidence. <30 trades is anecdotal, not statistical. Don't launch a strategy live with only 20 backtests: variance can fool you.

Overfitting Danger

Optimizing your strategy until it works perfectly on historical data is a trap. You're fitting to noise, not signal. Result: 95% win rate in backtest, 35% in real trading. Solution: Out-of-sample testing (data you didn't use for optimization).

Statistical Significance

A backtest with 60% win rate might be luck. Use confidence intervals. Ask: "What's the probability this result is chance?" If p-value < 0.05 (5%), you have statistical confidence. Otherwise, you need more data.