If X follows binomial distribution with parameters n = 9 and p such that P(X = 4) = 4P(X = 5), then P(X = 0) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\left(\displaystyle\frac{4}{5}\right)^{9}\)
For the binomial distribution with n = 9 and q = 1 − p,
\(P(X = k) = {}^{9}C_{k}\,p^{k}q^{9-k}\)
Since \({}^{9}C_{4} = {}^{9}C_{5}\), the condition P(X = 4) = 4P(X = 5) becomes
\(p^4q^5 = 4p^5q^4\)
\(q = 4p\)
\(1 - p = 4p\)
\( \displaystyle p = \frac{1}{5},\quad q = \frac{4}{5} \)
\( \displaystyle P(X = 0) = q^9 = \left(\frac{4}{5}\right)^{9} \)