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 $n\mathit{}=10$
Let p be the probability that a drawn egg is defective.

$$\begin{gathered} \therefore p=10\%=\frac{1}{10},q=\frac{9}{10} \\ \mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{10}C_r{\left(\frac{1}{10}\right)}^r{\left(\frac{9}{10}\right)}^{10-r},r=0,1,2....10 \\ P(\mathrm{there}\mathrm{is}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{defective}\mathrm{egg})=P(X\ge 1) \\ =1-P(X=0) \\ =1-{}^{10}C_0{\left(\frac{1}{10}\right)}^0{\left(\frac{9}{10}\right)}^{10-0} \\ =1-{\left(\frac{9}{10}\right)}^{10} \end{gathered}$$

$=1-\frac{9^{10}}{{10}^{10}}$