Conditional Probability — P(A
Probability
9
JEE Qs
8%
Hard
60
min
Master the art of precisely identifying the 'given' event to correctly define the reduced sample space and the intersection for accurate calculation of conditional probability.
🧮 Key Formulas
✅ Key Points for JEE
- 1Conditional probability means the probability of an event 'A' occurring, given that another event 'B' has already occurred.
- 2The condition 'given B' effectively reduces the sample space to event B, meaning P(B) becomes the new denominator, and P(B) must be greater than 0.
- 3Always carefully identify the conditioning event (the 'given' part) as it defines the reduced sample space.
- 4P(A|B) is generally not equal to P(B|A); the order of events matters significantly.
- 5Conditional probability satisfies all axioms of probability (e.g., 0 ≤ P(A|B) ≤ 1, P(S|B) = 1 where S is the sample space).
⚠️ Common Mistakes
- ✕Confusing P(A|B) with P(A ∩ B) or P(B|A).
- ✕Incorrectly identifying the conditioning event (the 'given' event), leading to a wrong denominator in the formula.
- ✕Assuming events are independent and using P(A ∩ B) = P(A) * P(B) when they are dependent, or vice-versa.
📝 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 Ch 13: Probability