A car manufacturing factory has two plants \(X\) and \(Y\). Plant \(X\) manufactures \(70 \%\) of cars and plant \(Y\) manufactures \(30 \%\) of cars. \(80 \%\) of cars at plant \(X\) and \(90 \%\) of cars at plant \(Y\) are rated as standard quality. A car is chosen at random and is found to be standard quality. The probability that it has come from plant \(X\) is

Let \(E=\) the event that the car is of standard quảlity.
Let \(A_{1}=\) the event that the car is manufactured in plant \(X\).
and \(A_{2}=\) the event that the car is manufactured in plant \(Y\).
Now, \(P\left(A_{1}\right)=\frac{70}{100}=\frac{7}{10}, P\left(A_{2}\right)=\frac{30}{100}=\frac{3}{10}\) \(P\left(\frac{E}{A_{1}}\right)=\) Probability that a standard quality car is manufactured in plant \(X=\frac{80}{100}=\frac{8}{10}\) \(P\left(\frac{E}{A_{2}}\right)=\) Probability that a standard quality car is manu factured in blant \(Y=\frac{90}{100}=\frac{9}{10}\) \(P\left(\frac{A_{1}}{E}\right)=\) Probability that a standard quality car has come from plant \(X\)
\(\begin{aligned}
=& \frac{P\left(A_{1}\right) \times P\left(\frac{E}{A_{1}}\right)}{P\left(A_{1}\right) \cdot P\left(\frac{E}{A_{1}}\right)+P\left(A_{2}\right) \cdot P\left(\frac{E}{A_{2}}\right)} \\
=& \frac{\frac{7}{10} \times \frac{8}{10}}{\frac{7}{10} \times \frac{8}{10}+\frac{3}{10} \times \frac{9}{10}} \\
=& \frac{7 \times 8}{7 \times 8+3 \times 9} \\
=& \frac{56}{56+27}=\frac{56}{83}
\end{aligned}\)
Hence, the required probability is \(\frac{56}{83}\).