Probability measures the likelihood of an event occurring and is a fundamental concept in statistics and real-world decision-making. It is represented as a number between 0 and 1, where 0 means an impossible event and 1 means a certain event.
\[ P(A) \in [0,1] \quad \text{(Probability ranges between 0 and 1)} \] \[ P(A') = 1 - P(A) \quad \text{(Complement Rule)} \]
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] \[ P(A \cap B) = P(A)P(B) \quad \text{(if A and B are independent)} \tag{1} \]
\[ P(A \cap B) = P(A|B)P(B) = P(B|A)P(A) \] \[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} \tag{2} \]