rag/knowledge_base/probability_basics.md

Probability Basics

Fundamental Definitions

• $P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$
• $0 \leq P(A) \leq 1$
• $P(A') = 1 - P(A)$ (complement)
• $P(S) = 1$ (sample space)

Addition Rule

• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• If A, B mutually exclusive: $P(A \cup B) = P(A) + P(B)$
• $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$

Conditional Probability

$P(A|B) = \frac{P(A \cap B)}{P(B)}$ where $P(B) > 0$

Multiplication Rule

• $P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$
• If A, B independent: $P(A \cap B) = P(A) \cdot P(B)$

Bayes' Theorem

$P(A_i|B) = \frac{P(B|A_i) \cdot P(A_i)}{\sum_{j} P(B|A_j) \cdot P(A_j)}$

Total Probability

$P(B) = \sum_{i} P(B|A_i) \cdot P(A_i)$ where $A_i$ form a partition of sample space

Independence

Events A, B are independent iff:

• $P(A \cap B) = P(A) \cdot P(B)$
• $P(A|B) = P(A)$
• $P(B|A) = P(B)$

JEE Tips

• Draw Venn diagrams for union/intersection problems
• Use complementary counting: $P(A) = 1 - P(A')$
• For "at least one" problems: $P(\text{at least one}) = 1 - P(\text{none})$