MathsMediumClass 12

Random Variable & Probability Distribution

Probability

JEE Qs

Hard

min

Master the definitions of random variable and probability distribution, and practice applying the formulas for mean and variance with attention to detail to avoid common calculation errors.

🧮 Key Formulas

P(X = x_i) >= 0 for all i

sum P(X = x_i) = 1 (for discrete random variables)

E(X) = sum (x_i * P(X = x_i)) (for discrete random variables)

E(X^2) = sum (x_i^2 * P(X = x_i)) (for discrete random variables)

Var(X) = E(X^2) - (E(X))^2

Var(X) = sum ((x_i - E(X))^2 * P(X = x_i)) (for discrete random variables)

Standard Deviation (SD) = sqrt(Var(X))

E(aX + b) = aE(X) + b

Var(aX + b) = a^2 Var(X)

✅ Key Points for JEE

1A random variable (RV) is a real-valued function whose domain is the sample space of a random experiment. It assigns a numerical value to each outcome.
2Probability distributions for discrete RVs (Probability Mass Function, PMF) assign probabilities P(X=x_i) to each possible value x_i such that P(X=x_i) >= 0 and sum P(X=x_i) = 1.
3Expectation E(X) represents the average or mean value of the random variable over many trials, and variance Var(X) measures the spread or dispersion of the values around the mean.
4Carefully distinguish between E(X^2) and (E(X))^2 when calculating variance; this is a common source of error.
5Understand how transformations E(aX+b) and Var(aX+b) affect expectation and variance respectively.

⚠️ Common Mistakes

✕Confusing the square of expectation (E(X))^2 with the expectation of the square E(X^2) when calculating variance.
✕Failing to ensure that the sum of probabilities for a discrete distribution equals 1, leading to incorrect calculations for expectation or variance.
✕Calculation errors in summations, especially with negative values or fractions, leading to incorrect mean or variance.

📝 Practice Questions

See all

Q5. 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

2025·MCQMedium

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

2025·MCQMedium

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

2025·MCQMedium

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

2025·MCQMedium

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

2025·MCQMedium

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

2025·MCQMedium

NCERT Chapters

Class 12 Maths Ch 13: Probability