Conditional ProbabilityFoundation

Concept

Conditional probability P(A|B) re-normalises: given that B happened, the sample space SHRINKS to B, and we ask what fraction of it is also A. Independence means the information 'B happened' changes nothing: P(A|B) = P(A), equivalently P(A∩B) = P(A)P(B). Mutually exclusive is the OPPOSITE of independent (for positive-probability events): knowing B occurred makes A impossible.

Key formula

Derivation

With 100 equally likely outcomes: P(A|B) = |A∩B|/|B| — count inside the reduced universe B.

Multiplication rule follows: P(A∩B) = P(B)·P(A|B) = P(A)·P(B|A). Chain it for three events: P(A∩B∩C) = P(A)P(B|A)P(C|A∩B).

Scenarios to explore

• Conditional Probability — Shrink the universe to B — independence vs exclusivity.

Real-world applications

• Card draws without replacement (probabilities update each draw).
• Reliability: P(system works | component failed).
• Weather forecasting: P(rain | humidity high).

JEE exam tips

• Venn + 100-outcome counting kills most conditional problems — avoid formula soup.
• Independence of A, B extends to complements: A ⊥ B ⟹ A ⊥ B̄, Ā ⊥ B, Ā ⊥ B̄.
• P(exactly one) = P(A) + P(B) − 2P(A∩B) — memorise, it recurs.

Common mistakes

• Confusing P(A|B) with P(B|A) — they differ unless P(A) = P(B).
• Treating mutually exclusive as independent — they're incompatible (unless a probability is 0).
• Forgetting to shrink the denominator to P(B).

Exam traps to avoid

• Pairwise independence of three events does NOT imply mutual independence.
• P(A|B) + P(Ā|B) = 1, but P(A|B) + P(A|B̄) is NOT generally 1.