Conditional Probability

Conditional probability measures the probability of an event occurring, given that another event has already occurred. It's a foundational concept in statistics and forms the mathematical basis for confidence in association rule mining.

The Formula

P(A|B) = P(A and B) / P(B)

This reads as "the probability of A given B" and is calculated by dividing the probability of both A and B occurring together by the probability of B occurring on its own.

A Simple Example

Suppose in a set of 100 customer transactions, 40 include both bread and butter, and 50 include bread overall. The probability of a customer buying butter given that they bought bread is:

P(Butter | Bread) = 40/100 / 50/100 = 0.8

Meaning there's an 80% chance a customer who bought bread also bought butter.

Connection to Association Rules

This is exactly how confidence is calculated in association rule mining - the confidence of a rule A -> B is nothing but the conditional probability P(B|A), the likelihood of B being purchased given that A was already purchased.

Independence

Two events are said to be independent if the occurrence of one doesn't affect the probability of the other - in that case, P(A|B) simply equals P(A).

Conditional probability is easy to misinterpret - P(A|B) is not the same as P(B|A); mixing these up is one of the most common statistical errors, sometimes called the "prosecutor's fallacy."

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