In the items produced by a factory, there are $10 \%$ defective items. A sample of 6 items is randomly chosen. Find the probability that this sample contains at least 3 defective items

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In the items produced by a factory, there are $10 \%$ defective items. A sample of 6 items is randomly chosen. Find the probability that this sample contains at least 3 defective items

The probability that the item is defective $=\frac{1}{10}=p$
The probability that the bulb will not fuse $=1-\frac{1}{10}=\frac{9}{10}=q$
Using Bernoulli’s we have,
$\begin{array}{l} P(\text { Success }=x)={ }^{n} C_{x} \cdot p^{x} \cdot q^{(n-x)} \\ x=0,1,2, \ldots \ldots . . n \text { and } q=(1-p), n=6 \end{array}$
The probability of at least 3 defective items are,
$\begin{array}{l} P(3)+P(4)+P(5)+P(6) \\ \Rightarrow{ }^{6} C_{3} \cdot\left(\frac{1}{10}\right)^{3}\left(\frac{9}{10}\right)^{3}+{ }^{6} C_{4} \cdot\left(\frac{1}{10}\right)^{4}\left(\frac{9}{10}\right)^{2}+{ }^{6} C_{5} \cdot\left(\frac{1}{10}\right)^{5}\left(\frac{9}{10}\right)^{1}+{ }^{6} C_{6} \cdot\left(\frac{1}{10}\right)^{6}\left(\frac{9}{10}\right)^{0} \end{array}$
$\Rightarrow \frac{15850}{10^{6}}$