Conditional probability formalises how the arrival of new information modifies the likelihood of an uncertain event.
When condition B occurs, all outcomes outside of B become impossible, shrinking the effective sample space to set B.
The conditional probability of A given B represents the fraction of probability mass within B that also belongs to A.
Mathematically, it is defined as the joint probability of A and B divided by the probability of B, provided B is non-zero.
If the conditioning probability is zero, the conditional probability is undefined.
Rearranging the definition yields the multiplication rule: joint probability equals conditional probability times conditioning probability.
For multi-step processes, the chain rule extends this multiplication across any sequence of consecutive conditions.
For example, drawing two cards without replacement conditions the second card draw on what was removed during the first draw.
The law of total probability uses conditional probabilities to evaluate an overall outcome across several mutually exclusive scenarios.
It computes the total probability of an event by weighting its conditional probabilities across every possible conditioning case.
Conditional probability explains why intuitive probabilistic reasoning often fails during medical and legal deliberations.
For instance, the probability of having a disease given a positive test is often surprisingly low for rare conditions.
This discrepancy occurs because false positives from the large healthy population outnumber true positives from the tiny sick group.
The prosecutor fallacy is another common legal error that confuses the probability of evidence given innocence with innocence given evidence.
Analysts must always distinguish clearly between the conditional probability of A given B and the reverse probability of B given A.