Independent Events
Probability
9
JEE Qs
8%
Hard
50
min
Always verify independence, either by applying the P(A ∩ B) = P(A)P(B) condition or by logically inferring it from the problem description, as it's a critical factor in solving complex probability problems.
🧮 Key Formulas
✅ Key Points for JEE
- 1Two events A and B are independent if and only if the occurrence of one does not affect the probability of the other's occurrence. This is mathematically verified by P(A ∩ B) = P(A)P(B).
- 2If A and B are independent events, then their complements A' and B' are also independent. Similarly, (A' and B) and (A and B') are independent.
- 3Do not confuse independent events with mutually exclusive events. Mutually exclusive events (A ∩ B = φ) can only be independent if at least one of them has a probability of zero.
- 4For problems involving 'at least one' event occurring among several independent events, it is often simpler to calculate the probability of 'none of them occurring' using complements and then subtract from 1.
- 5Events from experiments like drawing with replacement, tossing multiple coins/dice, or simultaneous independent trials are typically considered independent.
⚠️ Common Mistakes
- ✕Assuming events are independent without logical justification or explicit statement, especially when dealing with conditional probabilities.
- ✕Confusing independent events with mutually exclusive events and incorrectly using formulas like P(A ∪ B) = P(A) + P(B) for independent events or P(A ∩ B) = P(A)P(B) for mutually exclusive events.
- ✕Incorrectly applying the property P(A|B) = P(A) only when events are independent; otherwise, P(A|B) = P(A ∩ B) / P(B).
📝 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 Mathematics Ch 13: Probability