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Conditional Probability

Nearly every real probability is conditional. You are never estimating in a vacuum; you are estimating given everything you already know. When a large buy order hits your quote, the probability the asset is undervalued changes. Conditioning is the formal version of that update.

Conditional probability
P(A∣B)=P(A∩B)P(B)for P(B)>0P(A \mid B) = \frac{P(A \cap B)}{P(B)} \qquad \text{for } P(B) > 0

Undefined when B cannot happen, which is not a technicality: it is why the denominator has to be checked.

Read it as restricting the world to those outcomes where BB occurred, then asking what fraction of that smaller world also has AA. The denominator renormalises: BB is now the whole sample space.

Rearranged, it gives the general multiplication rule, which always holds:

P(A∩B)=P(B) P(A∣B)P(A \cap B) = P(B)\,P(A \mid B)

Independence is exactly the special case P(A∣B)=P(A)P(A \mid B) = P(A), where conditioning changes nothing.

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