Question

Companies B1, B2 and B3 produce 30%, 45% and 25% of the cars respectively. It is known that 2%, 3% and 2% of the cars produced from B1, B2 and B3 are defective. Find out i) the probability that a car purchased is defective? ii) If a car is purchased is found to be defective, find the probability that this car is produced by company B3?

Answer 1

Given :

%age of cars produced by company B1 = 30 %

%age of cars produced by company B2 = 45 %

%age of cars produced by company B3 = 25 %

%age of defective cars from B1 = 2%

%age of defective cars from B2 = 3%

%age of defective cars from B3 = 2%

To Find :

(i) Probability that a car purchased is defective

(ii) Probability that if a car is found to be defective it is produced by company B3

Solution :

Let the total no of cars produced be n .

So, the no of cars produced by B1 = $\frac{30}{100}\times n =\frac{3n}{10}$

No of cars produced by B2 = $\frac{45}{100}\times n = \frac{9n}{20}$

No of cars produced by B3 =$\frac{25}{100}\times n = \frac{n}{4}$

Now , the no of defective cars produced by B1 =$\frac{2}{100}\times \frac{3n}{10}= \frac{3n}{500}$

No of defective cars produced by B2 = $\frac{3}{100}\times\frac{ 9n}{20}= \frac{27n}{2000}$

No of defective cars produced by B3 =$\frac{2}{100}\times \frac{n}{4} = \frac{n}{200}$

So, the probability of purchasing a defective car is :

= total no of defective cars / total no of cars

= $\frac{(\frac{3n}{500} + \frac{27n}{2000} + \frac{n}{200})}{n}$

= $\frac{12n + 27n + 10n }{2000 n}$

= $\frac{49}{2000}$

And the probability that defective car is produced by B3 is :

= total no of defective cars manufactured by B3 / total no of defective cars

= $\frac{\frac{n}{200}}{\frac{3n}{500}+\frac{27n}{2000}+ \frac{n}{200}}$

=$\frac{\frac{n}{200}}{\frac{12n + 27n + 10n }{2000}}$

= $\frac{\frac{n}{200}}{\frac{49n}{2000}}$

=$\frac{n}{200}\times \frac{2000}{49n}$

= $\frac{10}{49}$

Hence ,

i) the probability that a car produce dis defective is =  $\frac{49}{2000}$

ii) the probability that a purchased defective car is produced  by B3 = $\frac{10}{49}$