Conditional Probability & Bayes' Rule

New information changes probability only relative to what was believed before. Conditional probability makes that update explicit and prevents a common mistake: confusing “the signal is common when the event is true” with “the event is probably true when the signal appears.”

Conditioning and Bayes' rule #

For events $A$ and $B$ with $P(B)>0$,

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

Bayes' rule reverses the direction:

$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B\mid A)P(A)+P(B\mid A^c)P(A^c)}.$$

$P(A)$ is the prior or base rate; $P(B\mid A)$ is the signal's likelihood when $A$ is true; and $P(A\mid B)$ is the updated or posterior probability.

A life example with 1,000 devices #

Suppose 1% of devices are defective. An alarm fires for 90% of defective devices and 5% of good ones. Among 1,000 devices, expect 10 defective, 9 true alarms, 990 good, and about 49.5 false alarms. Therefore

$$P(Defect\mid Alarm)=\frac{9}{9+49.5}\approx0.154.$$

The alarm is informative: probability rose from 1% to about 15.4%. But it did not make a defect more likely than not because false alarms act on a much larger base population.

Trading update #

Suppose a contract begins at $q=0.50$. Credible evidence raises the trader's estimate to $q=0.65$. That does not imply every venue quote should be copied: the evidence likelihood, base rate, time, and contract wording all belong in the update.

Application to no-hedge #

A reference VWAP is evidence $B$, not the target event $A$ itself. Treating $1-referenceVWAP$ as a fair value is defensible only when the reference and maker contracts are true complements with identical settlement semantics. A changed reference price should trigger a probability update; it is not a hedge and it does not remove $P(NO\mid B)$.