1.8 Definition vs. Theorem

What is the difference between a definition and a theorem? On pages 10 and 11 of the textbook, there is a rapid fire collection of pink boxes. We reproduce them here (notice that they may have different index numbers than the book – this live session book autoindexes and we’re not including every theorem and definition in this live session discussion guide).

Definition 1.2 Conditional Probability For \(A, B \in S\) with \(P(B) > 0\), the conditional probablity of \(A\) given \(B\) is \[P(A|B) = \frac{P(A\cap B)}{P(B)}.\]

Theorem 1.1 Multiplicative Law of Probability For \(A, B \in S\) with \(P(B) > 0\), \[P(A|B)P(B) = P(A \cap B)\]

Theorem 1.2 Baye’s Rule For \(A, B \in S\) with \(P(A) > 0\) and \(P(B) > 0\), \[P(A|B) = \frac{P(B|A)P(A)}{P(B)}.\]

• What would happen to the statement of the Multiplicative Law of Probability if we did not have the definition of Conditional Probability?
• How does one get from the definition, to the law?
• Can one get to Baye’s Rule wihtout using the Multiplicative Law of Probability?