Last Updated: May 2026
Probability is a guaranteed scoring chapter in JEE Main 2027 Mathematics — every paper carries 2-3 questions and the difficulty band is consistently moderate. It builds on Class 11 (sets, basic probability) and Class 12 (conditional probability, Bayes theorem, random variables, binomial distribution). This guide condenses every formula and 40 JEE-pattern questions with detailed solutions.
Probability — JEE Main Snapshot
| Parameter | Detail |
|---|---|
| NCERT Chapters | Class 11 Ch 16, Class 12 Ch 13 |
| JEE Main Questions / Year | 2-3 |
| Difficulty | Moderate |
| High-yield sub-topic | Bayes theorem, conditional probability |
1. Basic Definitions
- Sample Space (S): All possible outcomes.
- Event (E): Subset of S.
- Probability: P(E) = n(E)/n(S).
- Mutually Exclusive: E1 ∩ E2 = empty.
- Independent: P(E1 ∩ E2) = P(E1) × P(E2).
2. Conditional Probability
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
3. Total Probability and Bayes Theorem
P(A) = sum P(Eᵢ) × P(A|Eᵢ).
Bayes: P(Eₖ|A) = [P(Eₖ) × P(A|Eₖ)] / P(A).
4. Random Variables and Binomial Distribution
- Mean E(X) = sum xᵢ pᵢ.
- Variance = E(X²) − [E(X)]².
- Binomial: P(X = r) = C(n,r) p^r q^(n−r). Mean = np. Variance = npq.
5. Common JEE Tricks
- “At least one”: P(at least one) = 1 − P(none).
- Without replacement: use sequential conditional probabilities.
- Combinatorics inside probability: always check whether order matters.
40 Practice Questions — Sample of 10
Q1. Two cards from 52: probability both aces — (a) 1/221 (b) 1/26 (c) 1/13 (d) 4/52. Ans: (a)
Q2. P(A) = 1/2, P(B) = 1/3, P(A ∩ B) = 1/6 ⇒ P(A|B) = (a) 1/2 (b) 1/3 (c) 1/4 (d) 1/6. Ans: (a)
Q3. A and B independent, P(A) = 0.4, P(B) = 0.5. P(A ∪ B) = (a) 0.7 (b) 0.6 (c) 0.9 (d) 0.45. Ans: (a)
Q4. Two dice: probability sum 7 = (a) 1/12 (b) 1/9 (c) 1/6 (d) 7/36. Ans: (c)
Q5. Coin tossed 5 times: probability exactly 3 heads — (a) 5/32 (b) 10/32 (c) 16/32 (d) 1/32. Ans: (b)
Q6. Bag has 5 red, 7 black; two drawn: probability both red — (a) 5/33 (b) 7/22 (c) 5/22 (d) 7/33. Ans: (a)
Q7. Three machines (50%, 30%, 20%) defective rates (0.02, 0.03, 0.04). P(defective) = (a) 0.027 (b) 0.030 (c) 0.025 (d) 0.022. Ans: (a)
Q8. Continuing Q7, given defective, probability from machine 1 — (a) 10/27 (b) 9/27 (c) 8/27 (d) 5/27. Ans: (a)
Q9. Mean of binomial with n = 10, p = 1/5 is — (a) 1 (b) 2 (c) 3 (d) 4. Ans: (b)
Q10. A die rolled twice: probability same number both times — (a) 1/6 (b) 1/12 (c) 1/36 (d) 5/6. Ans: (a)
Common Pitfalls
- Mixing independence with mutual exclusivity.
- Ignoring “at least” reformulation via complement.
- Bayes setup errors — identify prior and likelihood first.
Frequently Asked Questions
How many probability questions in JEE Main 2027?
2-3 per shift, mostly conditional probability or binomial distribution.
Is Bayes theorem really tested?
Yes — at least one Bayes question features in 80% of JEE Main shifts.
Can I skip random variables and binomial?
No. Mean and variance of binomial appear in JEE Main almost annually.
Best book for probability?
NCERT Class 12 Chapter 13 plus problems from Cengage Probability or Arihant Skills in Mathematics.
How long for this chapter?
10-12 days for a first-pass with 60-80 problems plus revision.