Question

A company has two plants to manufacture televisions. The plant I manufacture $70$% of televisions and plant II manufacture $30$%. At plant I, $80$% of the televisions are rated as of standard quality and at plant II, $90$% of the televisions are rated as of standard quality. A television is chosen at random and is found to be of standard quality. The probability that it has come from plant II is

Answer 1

$E_1=$ Plant $1$ manufactures televisions.

$E_1=$ Plant $2$ manufactures televisions.

$P\left(E_1\right)=\frac{70}{100},P\left(E_2\right)=\frac{30}{100},P\left(\frac{A}{E_1}\right)=\frac{80}{100},P\left(\frac{A}{E_2}\right)=\frac{90}{100}$

By baye's theorem

$P\left(\frac{E_2}{A}\right)=\frac{P\left(E_2\right)P\left(\frac{A}{E_2}\right)}{P\left(E_2\right)P\left(\frac{A}{E_2}\right)+P\left(E_1\right)P\left(\frac{A}{E_1}\right)}=\frac{27}{27+56}=\frac{27}{83}$

$\therefore$ Probability it has came from plant $2$ is $\frac{27}{83}$