Binomial Distribution — Mean, variance
Probability
9
JEE Qs
8%
Hard
40
min
Master the identification of 'n' and 'p' in word problems, as these parameters are critical for applying the direct formulas for mean and variance.
🧮 Key Formulas
✅ Key Points for JEE
- 1Correctly identify 'n' (number of independent trials) and 'p' (probability of success in a single trial) from the problem statement.
- 2The sum of probabilities for a binomial distribution (q+p)^n always equals 1, where P(X=k) is the (k+1)th term in the expansion.
- 3Mean and variance formulas for binomial distribution (np and npq) are direct applications once n and p are known; derivation is not typically required in exams.
- 4Remember that 'q' is the probability of failure (1-p).
⚠️ Common Mistakes
- ✕Confusing 'p' (probability of success) with 'q' (probability of failure) or incorrectly identifying them in a context-based problem.
- ✕Using the general formulas for E(X) = Σx_i P(X=x_i) and Var(X) = Σx_i² P(X=x_i) - (E(X))² when the direct binomial formulas are much simpler.
- ✕Calculation errors, especially forgetting to take the square root for standard deviation or making arithmetic errors with 'p' and 'q'.
📝 Practice Questions
See allQ5. If A and B are two events such that P(A ∩B) = 0.1, and P(A ∣B) and P(B ∣A) are the roots of the equation – 12x2 −7x + 1 = 0, then the value of P(A∪B) is : P(A∩B) (1) 4 (2) 7 3 4 (3) 5 (4) 9 3 4
Q10.Let A = [aij] be a square matrix of order 2 with entries either 0 or 1 . Let E be the event that A is an invertible matrix. Then the probability P(E) is : 2025 (24 Jan Shift 2) JEE Main Previous Year Paper (1) 3 (2) 5 16 8 (3) 3 (4) 1 8 8
Q3. Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is m , n where gcd(m, n) = 1, then m + n is equal to : (1) 4 (2) 14 (3) 13 (4) 11
Q16.A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ2 denote the mean and variance of X , then the value of 64 (μ + σ2) is : (1) 51 (2) 64 (3) 32 (4) 48
Q2. One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is (1) 2 (2) 1 3 2 (3) 4 (4) 3 9 5
Q8. Two number k1 and k2 are randomly chosen from the set of natural numbers. Then, the probability that the value of ik1 + ik2, (i = √−1) is non-zero, equals ⎪ ⎪ 2025 (28 Jan Shift 1) JEE Main Previous Year Paper (1) 1 (2) 3 2 4 (3) 1 (4) 2 4 3
NCERT Chapters
- Class 12 Maths Part I Ch 13: Probability