Ajay is a vendor who informs that there are 2% rotten apples in the fruit bag. Find the probability that a sample of 10 apples will include not more than one rotten apple.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \left(\displaystyle\frac{98}{100}\right)^{9}\left(\frac{59}{50}\right) \)
The number of rotten apples X in a sample of 10 follows the binomial distribution with \( n=10 \), \( p=\frac{2}{100} \), \( q=\frac{98}{100} \). “Not more than one” means \( X\le 1 \):
\( \displaystyle P(X\le 1)=P(X=0)+P(X=1) \)
\( \displaystyle =\left(\frac{98}{100}\right)^{10}+{}^{10}C_{1}\left(\frac{2}{100}\right)\left(\frac{98}{100}\right)^{9} \)
Taking \( \left(\frac{98}{100}\right)^{9} \) out as a common factor:
\( \displaystyle P(X\le 1)=\left(\frac{98}{100}\right)^{9}\left(\frac{98}{100}+\frac{20}{100}\right) \)
\( \displaystyle =\left(\frac{98}{100}\right)^{9}\left(\frac{118}{100}\right) \)
\( \displaystyle =\left(\frac{98}{100}\right)^{9}\left(\frac{59}{50}\right) \)