Probability as quantified uncertainty

What probability is for

Probability is a language for talking about uncertainty quantitatively — that is, with numbers instead of vibes. 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 — a simulation of a strategy on past data — got a good Sharpe ratio (a score of return earned per unit of risk taken) by pure luck?
• How likely is a 20% drawdown — a 20% drop from your account's previous high point — 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. ("Frequentist" comes from frequency — how often something happens over many repeats.) 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. (Named after Thomas Bayes; here probability is a degree of belief, not a frequency.) This is the natural fit for one-off events — "will this strategy work?" or "will the Fed cut rates?"

Both are mathematically valid. Academic statistics mostly uses the frequentist view. Quant trading increasingly uses the Bayesian view, especially when sample sizes are small — when you have only a few observations to work with.

Three axioms — the entire foundation

Modern probability rests on three axioms — starting rules accepted without proof, from which everything else is derived (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 — events that cannot both happen at once — $P(A \text{ or } B) = P(A) + P(B)$.

In words (axiom 3): if A and B can never happen together, then the probability that A or B happens is just the probability of A plus the probability of B. Concrete example: a fair die shows a 1 with probability 1/6 and a 2 with probability 1/6. It can't show both at once, so the probability of "a 1 or a 2" is 1/6 + 1/6 = 2/6.

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 meet a probabilistic argument in the wild, you can check whether it respects these three rules.

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.

(To calibrate — to line up your stated probabilities with how often things actually happen. A well-calibrated forecaster's "70% confident" calls come true about 70% of the time.)

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 — a genuine statistical advantage. You need a sample: many trades, observed over time.

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 idea of treating an uncertain quantity (like a stock's return) as a number drawn from a distribution of possibilities. Predict:

• Suppose a machine picks a random number anywhere between 0 and 1, with infinite decimal places. What is the chance it picks exactly 0.500000... — not 0.500001, not 0.4999999, but exactly one half? (Hint: think carefully.)
• For a quantity that moves smoothly, like a price: is it more useful to ask "what is the chance the price is exactly 100.00?" or "what is the chance the price lands somewhere between 99 and 101?"