• Multiplication Rule: $P(A \;and\;B)=P(B) \cdot P(A|B)$

• Conditional Probability: $P(B|A)$ $=$ $\frac{P(A \;and\; B)}{P(A)}$

• Law of Total Probability: $P(A)=P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots+P(B_n)P(A|B_n)$

Combining all these equations we get Bayes' Rule:

$P(B|A)$ $=$ $\frac{P(A \;and\; B)}{P(A)}= \frac{P(B) \cdot P(A|B)}{P(A)}$

$=\frac{P(B) \cdot P(A|B)}{P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots+P(B_n)P(A|B_n)}$