MathsMediumClass 12

Probability — Bayes' Theorem + Distributions

Probability

JEE Qs

15%

Hard

min

For Bayes' Theorem, clearly define events and structure your solution by first computing the total probability of the observed event, and for binomial distribution, correctly identify n, p, and k parameters for accurate calculations.

🧮 Key Formulas

P(A_i|B) = [P(B|A_i) * P(A_i)] / sum(P(B|A_j) * P(A_j))

P(B) = sum(P(B|A_j) * P(A_j))

P(X=k) = C(n, k) * p^k * (1-p)^(n-k)

E(X) = np

Var(X) = np(1-p)

✅ Key Points for JEE

1Bayes' Theorem is used to find the posterior probability of a specific cause (A_i) given that an effect (B) has occurred.
2The denominator in Bayes' Theorem is the total probability of the effect (B), calculated using the Law of Total Probability over all possible causes.
3Binomial Distribution models the number of successes (k) in a fixed number of independent Bernoulli trials (n), where each trial has only two outcomes (success/failure) and the probability of success (p) is constant.
4Carefully identify the values of n (number of trials), p (probability of success), and k (number of successes) for binomial distribution problems.
5Remember that 'distributions' in JEE context primarily refers to the Binomial Distribution.

⚠️ Common Mistakes

✕Confusing P(A|B) with P(B|A) in problem statements and formula application.
✕Incorrectly identifying the mutually exclusive and exhaustive events {A_1, A_2, ..., A_n} that partition the sample space for the Law of Total Probability.
✕Mistakes in calculating combinations (C(n,k)) or powers (p^k, (1-p)^(n-k)) when applying the binomial probability mass function.
✕Failing to correctly interpret 'at least k successes', 'at most k successes', or 'exactly k successes' in binomial distribution problems.

📝 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 Mathematics Ch 13: Probability