Math · Probability

Conditional Probability

Conditioning changes the sample space. Once we know B happened, observations outside B are no longer relevant.

Concept

Start with the full population. Condition on B, then ask: among the observations that remain in B, what fraction are also in A? This is not the same as assuming A and B happen independently.

Equation

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

Focus or select a term to trace it in the population.

Manipulate

Adjust the two events and their overlap. Every cell is one equally likely observation.

Size of A Size of B Overlap A ∩ B

A onlyB onlyA ∩ BNeither

Live calculation

After conditioning, the numerator is the overlap and the denominator is B—not the whole population.

Independent is not mutually exclusive

Use the Independent preset.

$P (A | B) = P (A)$

$P (A \cap B) = P (A) P (B)$

Disjoint events have no overlap. When both events have nonzero probability, that makes them not independent.

A deck of cards

Let A be “the card is a King” and B be “the card is a face card.” There are 12 face cards, 4 of which are Kings.

$P (King | face card) = \frac{4}{12} = \frac{1}{3}$

Knowing the card is a face card changes the relevant set from 52 cards to 12.

Computing, data & machine learning

Conditional probability powers probabilistic models, classifiers, Bayesian inference, spam filtering, and reasoning under uncertainty. In testing contexts it helps describe how evidence changes a probability; it does not by itself make a medical conclusion.

Bayes’ Theorem

What if we know $P (B | A)$ , but want $P (A | B)$ ? That question leads naturally to Bayes’ Theorem.