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Conditional Probability

Formula

\[ P(A\mid B)=\frac{P(A,B)}{P(B)} \quad \text{for } P(B)>0 \]

Parameters

  • \(A,B\): events
  • \(P(A\mid B)\): probability of \(A\) after restricting to cases where \(B\) occurred

What it means

Conditional probability updates probabilities when you already know some event \(B\) happened.

What it's used for

  • Reasoning with evidence.
  • Factorizing joint probabilities.

Key properties

  • Product rule: \(P(A,B)=P(A\mid B)P(B)\).
  • If \(A\perp B\), then \(P(A\mid B)=P(A)\).

Common gotchas

  • Conditioning on rare events can dramatically change probabilities.
  • \(P(A\mid B)\) and \(P(B\mid A)\) are usually not equal.

Example

From a deck, let \(A=\) "card is a king", \(B=\) "card is a face card." Then \(P(A\mid B)=4/12=1/3\).

How to Compute (Pseudocode)

Input: event probabilities / joint distribution entries
Output: requested probability quantity

identify the relevant events/variables and required joint/marginal terms
apply the probability identity in the card formula
check denominator/normalization terms are valid (nonzero when required)
return the computed probability

Complexity

  • Time: Typically \(O(1)\) for a single event computation once required probabilities are available; larger table-based calculations scale with table size
  • Space: \(O(1)\) extra space for a single computation
  • Assumptions: Probability terms (joint/marginals/conditionals) are already known or computed separately

See also