If a box has 100 pens of which 10 are defective, then what is the probability that out of a sample of 5 pens drawn one by one with replacement atmost one is defective?

Here, $n = 5,p = \frac{{10}}{{100}} = \frac{1}{{10}}$ and $q = \frac{9}{{10}}$
$r \le 1$

$\Rightarrow$ $r = 0,1$

Also, $P(X = r){ = ^n}{C_r}{p^\prime }{q^{n - r}}$
$\therefore$ $P(X = r) = P(r = 0) + P(r = 1)$

${ = ^5}{C_0}{\left( {\frac{1}{{10}}} \right)^0}{\left( {\frac{9}{{10}}} \right)^5}{ + ^5}{C_1}{\left( {\frac{1}{{10}}} \right)^1}{\left( {\frac{9}{{10}}} \right)^4}$

$= {\left( {\frac{9}{{10}}} \right)^5} + 5 \cdot \frac{1}{{10}} \cdot {\left( {\frac{9}{{10}}} \right)^4}$

$= {\left( {\frac{9}{{10}}} \right)^5} + \frac{1}{2}{\left( {\frac{9}{{10}}} \right)^4}$

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