Question

A company has two plants to manufacture scooters. Plant-1 manufactures 70 of the scooters and plant-2 manufactures 30%. At plant-1, 30% of the scooters are rated of standard quality and at plant-2, 90% of the scooters are rated as standard quality. A scooter is chosen at random and is found to be of standard quality. Find the probability that it has come from plant $2$.

Answer 1

Consider the problem

Let,

$E_1$ is the plant $I$ manufactured scooter

$E_2$ the plant $II$ manufactured scooter

And, $A$ is scooter of standard quality

Then,

$\begin{array}{c}P\left(E_1\right)=\frac{70}{100}\\ P\left(E_2\right)=\frac{30}{100}\end{array}$

And,

$\begin{array}{c}P\left(\frac{A}{E_1}\right)=\frac{30}{100}\\ P\left(\frac{A}{E_2}\right)=\frac{90}{100}\end{array}$

Now, apply Bayes' Theorem

$P\left(\frac{E_2}{A}\right)=\frac{P\left(E_2\right)P\left(\frac{A}{E_2}\right)}{P\left(E_1\right)P\left(\frac{A}{E1}\right)+P\left(E_2\right)P\left(\frac{A}{E_2}\right)}$

$\begin{array}{c}=\frac{\left(\frac{30}{100}\times \frac{90}{100}\right)}{\left(\frac{70}{100}\times \frac{30}{100}\right)+\left(\frac{30}{100}\times \frac{90}{100}\right)}\\ =\frac{27}{21+27}\\ =\frac{27}{48}\end{array}$

Hence, The probality that the standard quality has come out from Plant $II$ is $\frac{27}{48}$