From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Explanation

It is given that out of `30` bulbs, `6` are defective.

`Rightarrow` Number of non-defective bulbs `=30-6=24`

`4` bulbs are drawn from the lot with replacement.

Let `{X}` be the random variable that denotes the number of defective bulbs in the selected bulbs.

`therefore {X}` can take the value` 0,1,2,3,4.`

`therefore {P}({X}=0)={P}(4` non-defective and `0` defective )`=^{4} C_{0} times frac{4}{5} times frac{4}{5} times frac{4}{5} times frac{4}{5}=frac{256}{625}`

`P(X=1)=P(3` non-defective and `1`defective )`=^{4} C_{1} times(frac{1}{5}) times(frac{4}{5})^{3}=frac{256}{625}`

`P(X=2)=P(2` non-defective and `2`defective )`=^{4} C_{2} times(frac{1}{5})^{2} times(frac{4}{5})^{2}=frac{96}{625}`

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