Conditional probability and independence

The question conditioning answers

Every trading idea of the form "when X happens, Y tends to follow" is a claim about conditional probability: the probability of $A$ given that $B$ happened.

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Read it as bookkeeping, not magic: throw away every outcome where $B$ didn't happen, then ask how often $A$ occurs in what's left. You're re-normalizing probabilities to a smaller sample space.

Concrete example. Suppose over 5,000 trading days, 2,300 were down days ($P(\text{down}) = 0.46$). Now look only at the 2,300 days that followed a down day, and suppose 1,081 of them were also down. Then:

$P(\text{down today} \mid \text{down yesterday}) = \frac{1081}{2300} = 0.47$

Yesterday's fall moved today's odds from 46% to 47% — almost nothing. This is roughly what real daily index data looks like: the sign of yesterday's return tells you very little about the sign of today's. (Magnitudes are another story — hold that thought.)

Independence

Two events are independent when conditioning changes nothing:

$P(A \mid B) = P(A) \quad \Longleftrightarrow \quad P(A \cap B) = P(A)\,P(B)$

The product form is the standard definition (it works even when $P(B) = 0$), but the conditional form is the intuition: knowing $B$ happened teaches you nothing about $A$.

Two things developers often get backwards:

• Independence is an empirical property, not a default. Whether daily returns are independent is a fact about markets you must check, not an axiom you get for free.
• Independent ≠ mutually exclusive. Mutually exclusive events are maximally dependent: if one happened, the other's probability drops to zero.

Why the √252 rule is secretly an independence assumption

In A1-04 and A2-02 you annualized volatility by multiplying daily volatility by $\sqrt{252}$. That rested on variances adding across days:

$\text{Var}(r_1 + r_2 + \cdots + r_{252}) = \sum_t \text{Var}(r_t) \quad \text{only if the covariance terms are zero}$

Independence is what kills those cross terms. If returns were positively autocorrelated (down days breeding down days), true annual volatility would be larger than $\sigma_d \sqrt{252}$ — the rule would understate risk.

The honest empirical picture for daily equity index returns:

• Return signs and levels: close to independent. Autocorrelation of daily returns is near zero. This is why the √252 rule works as well as it does.
• Return magnitudes: emphatically NOT independent. Big-move days follow big-move days — the volatility clustering from A2-02. Squared returns are strongly autocorrelated even when returns themselves aren't.

So "returns are independent" is a decent approximation for scaling volatility across time, and a terrible one for predicting how volatile tomorrow will be.

The inversion trap

The single most common conditional-probability error is confusing $P(A \mid B)$ with $P(B \mid A)$. They can be wildly different, because each conditions on a different base rate.

Markets version: nearly every crash was preceded by elevated volatility, so $P(\text{high vol} \mid \text{crash})$ is close to 1. Does high volatility mean a crash is coming? That's the other conditional, $P(\text{crash} \mid \text{high vol})$ — and it's small, because high-vol periods are common and crashes are rare. The rarity of crashes (the base rate) drags the inverted probability down.

Every "this indicator preceded the last 5 crashes" pitch you will ever read commits this inversion. The next lesson (Bayes' theorem) gives you the formula that converts one conditional into the other — and the base rate is the price of conversion.

Conditional is not causal

$P(A \mid B) > P(A)$ says the events co-occur more than chance. It does not say $B$ causes $A$. Conditioning on "the VIX is high" raises the probability of a turbulent week ahead, but the VIX causes nothing — it's a barometer, not the weather. Both are driven by the same underlying stress. Statistical dependence is symmetric ($A$ informs $B$ exactly as much as $B$ informs $A$); causation is not.

For trading this distinction is surprisingly forgiving in one way and brutal in another: you can profit from a stable conditional relationship without understanding its cause — but relationships without a causal anchor are exactly the ones that vanish when the regime changes.

Explain it back

Without scrolling up, explain to an imaginary colleague in 2–3 sentences: what does $P(A \mid B)$ mean mechanically, and in what specific sense are daily stock returns independent versus not independent?