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 that one event happens, given that another event already happened.

The notation is new, so read it slowly:

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

In words: "the probability of A, given that B happened, equals the probability that A and B both happen, divided by the probability of B." Two new symbols here. The vertical bar $\mid$ is read "given that" — everything to its right is the condition you're assuming. The symbol $\cap$ is read "and": $P(A \cap B)$ is the probability that both events happen together.

Read the formula as bookkeeping, not magic. Throw away every outcome where $B$ didn't happen. Then ask: how often does $A$ occur in what's left? You're re-normalizing — rescaling the remaining probabilities so they add up to 1 again — inside a smaller sample space (the shrunken set of outcomes that survived the filter).

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:

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

In words: the probability of a down day, given that yesterday was down, is 1,081 out of 2,300 — about 0.47. Notice the recipe: keep only the days where the condition held, then count.

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 (up or down) tells you very little about the sign of today's. (Magnitudes — how big the moves are — are another story. Hold that thought.)

Independence

Two events are independent when conditioning changes nothing:

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

In words: the left side says "the probability of A given B is just the plain probability of A" — learning that B happened moved nothing. The double arrow $\Longleftrightarrow$ is read "is equivalent to": whenever one side is true, so is the other. The right side says "the probability that both happen equals the two probabilities multiplied together." Quick check with a fair coin and a fair die: $P(\text{heads and rolling a 6}) = 0.5 \times 1/6 = 1/12$, because the coin can't influence the die.

The product form (the right side) is the standard definition — it works even when $P(B) = 0$. But the conditional form (the left side) 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:

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

In words: the variance of the whole year's return — all 252 daily returns added together — equals each day's variance summed up in a loop, one term per day. But that clean sum only holds if every covariance term is zero. (Covariance, from A1-04, is the number measuring how two returns move together.) Independence is what kills those cross terms.

If returns were positively autocorrelated — autocorrelation is the correlation of a series with its own past, so here: down days breeding more down days — the cross terms would be positive. True annual volatility would then be larger than $\sigma_d \sqrt{252}$, and the rule would understate risk.

The honest empirical picture for daily equity index returns:

• Return signs and levels: close to independent. The 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 $P(B \mid A)$ — "A given B" versus "B given A." They can be wildly different, because each one conditions on a different base rate — how common the conditioning event is overall, before you know anything else.

Markets version. Nearly every crash was preceded by elevated volatility, so $P(\text{high vol} \mid \text{crash})$ — the probability that volatility was high, given that a crash happened — 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. Why? High-vol periods are common; 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

$P(A \mid B) > P(A)$ — read "the probability of A given B is greater than the plain probability of A" — says the events co-occur more often than chance. It does not say $B$ causes $A$. Conditioning on "the VIX is high" (the VIX is a market index that measures expected volatility) 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 symmetric.

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 — the prevailing market environment — 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?