Let X~Bin (3, p) be a binomial random variable. If \(P(X=3)=\displaystyle\frac{1}{12}P(X=1)\), then p is equal to :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{3}\)
For a binomial variable with n = 3, \(P(X = r) = {}^{3}C_{r}\,p^r q^{3-r}\), where q = 1 − p.
\(P(X = 3) = p^3\)
\(P(X = 1) = 3pq^2\)
The given condition:
\( \displaystyle p^3 = \frac{1}{12} \times 3pq^2 = \frac{pq^2}{4} \)
Dividing by p (p ≠ 0):
\(4p^2 = q^2\)
Since p and q are positive:
\(q = 2p\)
\(1 - p = 2p\)
\( \displaystyle p = \frac{1}{3} \)