JEE Main 2022 · 26 July · morning · Single correct · +4 / −1
The mean and variance of a binomial distribution are α and α/3 respectively. If P(X=1)=4/243, then P(X=4 or 5) is equal to :
- (a) 5/9
- (b) 64/81
- (c) 16/27
- (d) 145/243
Answer and solution
Answer (c) 16/27
Solution
Maths sir · 3 steps
Divide variance by mean to get q at once. Then the given P(X = 1) tells you n.
- Find p and q. Mean is np = α and variance is npq = α/3. Dividing gives q = 1/3, so p = 2/3.
- Find n. P(X=1) = n · 2/3 · (1/3)ⁿ⁻¹ = 2n/3ⁿ. Setting this equal to 4/243 gives n/3ⁿ = 2/243 = 6/729, so n = 6.
P(X=4) + P(X=5) = C(6, 4)2⁴/3⁶ + C(6, 5)2⁵/3⁶ = (240 + 192)/729
- So. 432/729 = 16/27, which is option (c).
The idea to keep. In a binomial question, variance divided by mean is q. Find q first, then use any one given probability to fix n.
Answer from the NTA final answer key. Solution by PrepWiser.