The mean and variance of a Binomial distribution are 4 and \(\displaystyle\frac{4}{3}\) respectively, then the value of \(\mathrm{P}(\mathrm{X}\ge 1)\) is :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{728}{729}\)
For a binomial distribution, mean = np and variance = npq:
\(np = 4\)
\(npq = \frac{4}{3}\)
Dividing the second by the first:
\(q = \frac{1}{3},\quad p = 1 - q = \frac{2}{3}\)
\(n = \frac{4}{p} = 6\)
Using the complement:
\(P(X \ge 1) = 1 - P(X = 0) = 1 - q^6\)
\( \displaystyle = 1 - \left(\frac{1}{3}\right)^6 = 1 - \frac{1}{729} \)
\( \displaystyle = \frac{728}{729} \)