If a random variable X follows binomial distribution with mean \( \displaystyle\frac{4}{3} \) and variance \( \displaystyle\frac{8}{9} \). Which of the following are correct?
(A) Probability of success is \( \displaystyle\frac{1}{3} \).
(B) Probability of failure is \( \displaystyle\frac{2}{3} \).
(C) Number of trials is 4.
(D) \( P(X\ge 1)=\displaystyle\frac{35}{81} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
For a binomial distribution, mean = np and variance = npq. Dividing the variance by the mean:
\( \displaystyle q=\frac{npq}{np}=\frac{8/9}{4/3} \)
\( \displaystyle q=\frac{2}{3} \)
\( \displaystyle p=1-q=\frac{1}{3} \)
\( \displaystyle n=\frac{np}{p}=\frac{4/3}{1/3} \)
\( \displaystyle n=4 \)
\( \displaystyle P(X\ge 1)=1-P(X=0)=1-q^{4} \)
\( \displaystyle =1-\frac{16}{81}=\frac{65}{81} \)
So \( p=\frac{1}{3} \), \( q=\frac{2}{3} \), \( n=4 \) and \( P(X\ge 1)=\frac{65}{81} \).
Checking the statements
(A) Probability of success is \( \frac{1}{3} \). — Correct. \( p=\frac{1}{3} \).
(B) Probability of failure is \( \frac{2}{3} \). — Correct. \( q=\frac{2}{3} \).
(C) Number of trials is 4. — Correct. \( n=4 \).
(D) \( P(X\ge 1)=\frac{35}{81} \) — Incorrect. \( P(X\ge 1)=\frac{65}{81} \), not \( \frac{35}{81} \).