2.4 Conditional Probability

In data science we are often interested in the relationship between two or more events.

Question: If \(A\) has happened, what is the probability that \(B\) also happens? Are they correlated?

Definition: Assume \(\mathbb{P}[B] \neq 0\), then the conditional probability of \(A\) given \(B\) is: \[ \mathbb{P}[A \mid B ] = \frac{\mathbb{P}[A \cap B]}{\mathbb{P}[B]} \]

This is a definition. There are alternative formulations that take this as foundational instead, see for example E.T. Jaynes “Probability Theory, The Logic of Science”

Conditional probabilities obey the axioms and are legitamate probabilities as shown in text.