Product Rule (Probability)
Definition
The rule that P(A and B) = P(A) x P(B) holds only when A and B are independent. Applying it to correlated events produces probabilities that are far smaller than the true values, making a coincidence appear impossibly rare.
- Rule
- P(A and B) = P(A) x P(B)
- Requirement
- Statistical independence
- Failure mode
- Applying it to correlated events
- Consequence
- Artificially tiny, misleading probabilities
Common questions
Why is the product rule a recurring source of error in forensic statistics?+
Analysts sometimes multiply probabilities of several traits or events together to get a headline rarity figure without first checking whether the traits are actually independent. When they are correlated, as many physical or behavioural traits are, the multiplication understates the true probability, exaggerating the rarity of the coincidence.
Which historical courtroom error is most associated with misuse of the product rule?+
Cases involving multiplied trait probabilities presented without empirical justification for independence, such as the US People v. Collins case, are commonly cited as illustrations of what happens when the product rule is applied to non-independent events.
Related terms
- Defence Fallacy
- The converse error of inflating the importance of the RMP by arguing that, because many people in the population share the profile,...
- Likelihood Ratio (LR)
- The ratio of two conditional probabilities: the probability of the observed evidence given the prosecution's hypothesis (same source), divided by the probability...
- Prosecutor's Fallacy
- The error of treating the RMP (or its reciprocal) as the probability that the defendant is innocent, or as the probability that...
- Reference Class
- The population against which a probability or frequency is calculated. Choosing the wrong reference class, for example using general population allele frequencies...
- Transposition of the Conditional
- The mathematical name for the error at the core of the prosecutor's fallacy. P(A | B) and P(B | A) are generally...