Probability as quantified uncertainty

What probability is for

Probability is a language for talking about uncertainty quantitatively — without it you can only say "maybe" or "I don't know." With it, you can say "I'm 70% confident," "this risk is worth $200 per trade," "this strategy makes sense to size at 2% of capital."

In trading, almost everything you care about is uncertain:

• Will this stock close up tomorrow?
• What's the chance my backtest's Sharpe was lucky?
• How likely is a 20% drawdown in the next year?

Without probability, these questions have no honest answers. With it, they become tractable.

Two views of probability (you need both)

Frequentist view

A probability of 0.7 means: in many repeated trials of this situation, the event happens 70% of the time. This is the natural fit for things you observe many times — daily returns, dice rolls, sample averages.

Bayesian view

A probability of 0.7 means: given everything I know, I believe the event will happen with 70% confidence. This is the natural fit for one-off events — "will this strategy work?" or "will the Fed cut rates?"

Both are mathematically valid. The frequentist approach dominates academic statistics; the Bayesian approach is increasingly common in quant trading, especially when sample sizes are small.

Three axioms — the entire foundation

Modern probability rests on three axioms (Kolmogorov, 1933):

1. Non-negativity. Every probability is between 0 and 1.
2. Total mass. The probability of something happening is 1.
3. Additivity. For disjoint events, $P(A \text{ or } B) = P(A) + P(B)$.

That's it. Every theorem follows from these three. We won't prove anything formally in this curriculum — but knowing the foundation exists is useful when you encounter probabilistic arguments and want to check whether they're tight.

The most useful intuition for trading

If you remember nothing else from this lesson, remember this:

Probabilities calibrate expectations, but a single outcome tells you almost nothing about whether your probability was right.

If I say "30% chance of rain" and it rains, I wasn't wrong. If I say "99% chance" and it doesn't rain, I might have been wrong — but a single rainless day doesn't prove it. You need many forecasts and many outcomes to evaluate a forecaster. The same logic applies to a trading strategy: one losing trade doesn't disprove an edge; you need a sample.

This is why we'll measure strategies over years and hundreds of trades, not weeks.

Predict before the next lesson

Tomorrow we introduce random variables — the abstraction that lets us talk about quantities (like a stock's return) as draws from a probability distribution. Predict:

• If you draw a random number uniformly between 0 and 1, what's the chance it's exactly 0.5? (Hint: think carefully.)
• For continuous quantities (like a price), does it make more sense to talk about $P(X = x)$ or $P(a \le X \le b)$?