Question

A company has two plants to manufacture television. Plant $1$ manufactures $70\%$ of televisions and plant $2$ manufactures $30\%$. At plant $1$, $80\%$ of the televisions are rated as of standard quality and at plant $2$, $90\%$ 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 $2$ is:

• A. $\frac{27}{83}$
• B. $\frac{29}{83}$
• C. $\frac{31}{83}$
• D. $\frac{35}{83}$

Answer 1

The correct option is A $\frac{27}{83}$

Let $E_1,E_2$ be the events that the T.V is manufactured at plant $1$, plant $2$ respectively.
Let '$E$' be the event that the TV is found to be of standard quality.
$P\left(E_1\right)=\frac{70}{100},P\left(E_2\right)=\frac{30}{100}$
$P\left(\frac{A}{E_1}\right)=\frac{80}{100}\text{and}P\left(\frac{A}{E_2}\right)=\frac{90}{100}$
$P\left(E_2/A\right)=\left(\frac{P\left(E_2\right).P\left(A/E_2\right)}{P\left(E_1\right).P\left(A/E_1\right)+P\left(E_2\right).P\left(A/E_2\right)}\right)=\frac{(\frac{30}{100}\times \frac{90}{100})}{(\frac{70}{100}\times \frac{80}{100})+(\frac{30}{100}\times \frac{90}{100})}=\frac{27}{56+27}=\frac{27}{83}$