Conditional Probability

• The meaning of P(A|B) — probability of A given that B has occurred.
• The formula P(A|B) = P(A ∩ B)/P(B) when P(B) > 0.
• Multiplication theorem: P(A ∩ B) = P(B)·P(A|B) = P(A)·P(B|A).
• Independent events: P(A|B) = P(A) when B does not affect A.
• To solve NCERT and JEE problems on cards, dice, and conditional selection without replacement.

Level 1 — Foundations

Verbal: Conditional probability updates our belief about event A once we know B has happened. The sample space shrinks to outcomes favourable to B.

Formula: P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.

Interpretation: Among outcomes where B occurs, what fraction also satisfies A?

Example: Draw one card from a deck. B = "red", A = "king". P(A|B) = 2/26 = 1/13 (two red kings among 26 red cards).

Complement: P(A′|B) = 1 − P(A|B) within the restricted sample space.

Level 2 — JEE / NEET depth

Multiplication rule: P(A ∩ B) = P(B)·P(A|B). For three events: P(A∩B∩C) = P(A)·P(B|A)·P(C|A∩B).

Independence: A and B independent iff P(A∩B) = P(A)P(B) iff P(A|B) = P(A).

Without replacement: Successive draws are dependent — update probabilities after each draw (hypergeometric flavour).

Law of total probability: If B₁,…,Bₙ partition sample space, P(A) = Σ P(Bᵢ)P(A|Bᵢ).

Tree diagrams: Branch probabilities multiply along paths; sum paths for combined events — excellent for JEE word problems.

Common setups: Two draws from urn; selecting students by gender then grade; defective items in manufacturing batches.

Conditional probability with cards

From a standard deck, given the card is a face card, find P(it is a Queen).

Step 1 — B = face card: 12 cards (J, Q, K in four suits).
Step 2 — A ∩ B = Queen and face = 4 Queens.
Step 3 — P(A|B) = 4/12 = 1/3.
Step 4 — Compare unconditional P(Queen) = 4/52 = 1/13 — conditioning on face card increases probability.

Multiplication theorem — two draws without replacement

Bag: 3 red, 2 blue balls. Draw 2 without replacement. Find P(both red).

Step 1 — P(R₁) = 3/5.
Step 2 — Given first red, 2 red left among 4: P(R₂|R₁) = 2/4 = 1/2.
Step 3 — P(R₁ ∩ R₂) = (3/5)(1/2) = 3/10.
Step 4 — Not independent: P(R₂) = 3/5 only with replacement.

• If P(A)=0.4, P(B)=0.5, P(A∩B)=0.2, find P(A|B).
• Two coins tossed. Given at least one head, P(both heads)?
• Are independent events always mutually exclusive?
• A family has two children. Given eldest is girl, P(both girls)?
• Stretch: Derive P(A|B) + P(A′|B) = 1 from definition.

NCERT Chapter 13 link: Conditional probability introduces updated sample spaces — draw tree diagrams for multi-stage experiments. NCERT Examples 13.3–13.5 illustrate card and urn problems; replicate method not just answers.

Exam connections: "Given that" phrasing always signals conditioning — denominator is P(given event), not P(A). JEE mixes conditional probability with Bayes in same paper — identify which tool first. Independence check: verify P(A∩B) = P(A)P(B) numerically when in doubt.

Study strategy: For without replacement, treat each stage as new reduced sample space. Write P(A∩B) = P(B)P(A|B) explicitly before substituting numbers — avoids inversion errors. Venn diagram shading helps visualise P(A|B) as fraction of B region inside A.

When studying Conditional Probability within Probability, start by listing every formula and definition on one page without looking at the textbook. Compare your list to NCERT — missing items indicate gaps to fix immediately. Work through at least two NCERT Examples for this section with steps written in full; examiners award method marks even when arithmetic slips.

For board exams (CBSE), long answers benefit from a clear structure: definition → explanation → diagram or formula → example → brief conclusion. Underline key terms. For JEE Main and NEET, prioritise conceptual traps and quick calculation paths; timed mixed quizzes of 10 questions after revision simulate exam pressure.

Cross-topic link: Coordinate geometry and vectors often combine with matrices; calculus links to physics kinematics problems.

Spaced revision: Review this note at 1 day, 3 days, and 7 days after first study. Attempt the Quick check questions closed-book, then open the Practice tab for graded reinforcement. Maintain an error log — repeated mistake patterns reveal whether the issue is concept, formula recall, or careless reading.

Diagram and terminology drill: For Mathematics, redraw key figures from memory and define every labelled part in one sentence. Vocabulary precision prevents mark loss in descriptive answers — use NCERT terms exactly as printed in the textbook.

Revision tip: Link this topic to adjacent Class 12 chapters before attempting mixed practice.

Open the Practice tab for graded questions on Conditional Probability.