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Binomial Distribution — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinomial Distribution3 Marks
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.
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Answer

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.
$\therefore\text{p}=10\%=\frac{1}{10},\text{q}=\frac{9}{10}$
Hence, the distribution is given by
$\text{P(X = r})=\text{ }^{10}\text{C}_{\text{r}}\big(\frac{1}{10}\big)^{\text{r}}\big(\frac{9}{10}\big)^{10-\text{r}},\text{r}=0,1,2\dots10$
$\text{P(there is at least one defectiv egg})=\text{P(X}\geq1)$
$=1-\text{P(X}=0)$
$=1-\text{ }^{10}\text{C}_0\big(\frac{1}{10}\big)^0\big(\frac{9}{10}\big)^{10-0}$
$=1-\big(\frac{9}{10}\big)^{10}$
$=1-\frac{9^{10}}{10^{10}}$

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