The probability that a person is a singer is 0.25. The probability that out of 8 persons 6 are singers is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{63}{2^{14}}\)
The number of singers X among 8 persons follows a binomial distribution with n = 8, \(p = \frac{1}{4}\) and \(q = \frac{3}{4}\):
\( \displaystyle P(X = 6) = {}^{8}C_{6}\left(\frac{1}{4}\right)^6\left(\frac{3}{4}\right)^2 \)
\( \displaystyle = 28 \times \frac{1}{4^6} \times \frac{9}{4^2} \)
\( \displaystyle = \frac{252}{4^8} = \frac{252}{2^{16}} \)
Dividing the numerator and the denominator by 4:
\( \displaystyle P(X = 6) = \frac{63}{2^{14}} \)