Question

A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that :

(i) all 10 arc defective

(ii) all 10 are good

(iii) at least one is defective

(iv) None is defective

Answer 1

We have,
A box containing 100 bulbs, out of which 20 are defective
$\therefore$ Number of good bulbs 100 - 20 = 80
Now, 10 balls are selected from inspection
$\therefore$ Numbers of elementary events in sample space

$n\left(S\right){=}^{100}{C}_{10}$

(i) Let E be the event that all 10 bulbs selected are defective

$n\left(E\right){=}^{20}{C}_{10}$

$\therefore P\left(E\right)=\frac{{}^{20}{C}_{10}}{{}^{100}{C}_{10}}$

(ii) Let E be the event that an 10 good bulbs arc selected

$\therefore n\left(E\right){=}^{80}{C}_{10}$

$\therefore P\left(E\right)=\frac{{}^{80}{C}_{10}}{{}^{100}{C}_{10}}$

(iii) Let E be the event that atleast one bulbs is defective
E= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10)

Where,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are numbers of defective bulbs
E be the event that none of the bulbs are defective

$\because \hat{E}$ be the event that none of the bulbs are defective

$\therefore n\left(\hat{E}\right)=\frac{{}^{80}{C}_{10}}{{}^{100}{C}_{10}}$

$\therefore P\left(E\right)=1-P\left(\hat{E}\right)$

$=1-\frac{{}^{80}{C}_{10}}{{}^{100}{C}_{10}}$

(iv) Let E be the event that none of the selected bulbs is defective, that is all bulbs are good.

So, $n\left(E\right){=}^{80}{C}_{10}$

$P\left(E\right)=\frac{{}^{80}{C}_{10}}{{}^{100}{C}_{10}}$