Question

Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.

Answer 1

Let X be the number of defective eggs drawn from 10 eggs.
Then, X follows a binomial distribution with \[n = 10\] 

Let p be the probability that a drawn egg is defective.

\[\therefore p = 10\] % = \[\frac{1}{10} , q = \frac{9}{10}\]
\[\text{ Hence, the distribution is given by }  \]
\[P(X = r) = ^{10}{}{C}_r \left( \frac{1}{10} \right)^r \left( \frac{9}{10} \right)^{10 - r} , r = 0, 1, 2 . . . . 10\]
\[P(\text{ there is at least one defective egg } ) = P(X \geq 1) \]
\[ = 1 - P(X = 0) \]
\[ = 1 -^{10}{}{C}_0 \left( \frac{1}{10} \right)^0 \left( \frac{9}{10} \right)^{10 - 0} \]
\[ = 1 - \left( \frac{9}{10} \right)^{10}\]

\[= 1 - \frac{9^{10}}{{10}^{10}}\]