Poisson Distribution
Probability
9
JEE Qs
8%
Hard
50
min
Master the conditions for applying Poisson distribution and its approximation to Binomial distribution, as well as the calculation of the parameter λ based on the problem context.
🧮 Key Formulas
✅ Key Points for JEE
- 1Poisson distribution models the number of times an event occurs in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event.
- 2It is used for 'rare events' where the number of trials (n) is very large, and the probability of success (p) in each trial is very small, but the average number of successes (np = λ) is finite.
- 3A unique property of the Poisson distribution is that its mean is equal to its variance (E(X) = Var(X) = λ).
- 4The sum of independent Poisson random variables is also a Poisson random variable. If X1 ~ P(λ1) and X2 ~ P(λ2), then X1 + X2 ~ P(λ1 + λ2).
- 5Poisson distribution can be used as an approximation to the Binomial distribution B(n, p) when n is large (n -> ∞) and p is small (p -> 0) such that their product np approaches a finite constant λ (np ≈ λ).
⚠️ Common Mistakes
- ✕Incorrectly identifying 'λ' (mean rate) when the time/space interval changes (e.g., if λ is given per hour, but probability is asked for a 3-hour interval, then the new λ should be 3 times the original).
- ✕Confusing the conditions for applying Poisson distribution versus Binomial distribution.
- ✕Errors in calculating probabilities involving 'at least' or 'at most' (e.g., P(X >= k) = 1 - P(X < k)).
📝 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
- Not directly covered in NCERT Class 11 or 12 Mathematics curriculum, but builds upon fundamental probability concepts.