Question

A box p has 1000 items of which 100 are defective. another box q has 500 items of which 20 are defective. the items of both the boxes are mixed and one items is randomly taken out. it is found to be defective. what is the probability that the items belongs to box p

Answer 1

Final Answer :
$\red { \boxed { \blue { \frac{5}{6} } }}$

Steps:
1) Box P - 100 Defective out of 1000
Box Q : 20 Defective out of 500 .

Now, items of both boxes are mixed.
Probability that the item selected is from Box P :
$= \frac{1000}{1000 + 500} = \frac{2}{3}$

Probability that the item is selected from Box Q :
$1- \frac{2}{3} = \frac{1}{3}$

2) Probability of selecting defective item from Box P :
$\frac{100}{1000} = \frac{1}{10}$

Probability of selecting defective item from Box Q :
$\frac{20}{500} = \frac{1}{25}$

3) Now, By Bayes Theorem,

Required Probability i . e :
Probability that item is from Box P when it is found defective already :
$= > \frac{ \frac{2}{3} \times \frac{1}{10} }{ \frac{2}{3} \times \frac{1}{10} + \frac{1}{3} \times \frac{1}{25} } \\ = > \frac{ \frac{1}{5} }{ \frac{1}{5} + \frac{1}{25} } \\ = > \frac{5}{6}$
Hence, Desired Probability is 5/6 .