There are two boxes, namely box-I and box-II. Box-I contains $\displaystyle 3$ red and $\displaystyle 6$ black balls. Box-II contains $\displaystyle 5$ red and $\displaystyle 5$ black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II. Case Study Based Question

\[\begin{aligned} & E_{1}: \text { Box I is selected } \\ & E_{2}: \text { Box II is selected } \\ & A: \text { A red ball is drawn from the selected bag } \\ & P\left(E_{1}\right)=P\left(E_{2}\right)=\frac{1}{2} \\ & P\left(A \mid E_{1}\right)=\frac{3}{9}=\frac{1}{3}, P\left(A \mid E_{2}\right)=\frac{5}{10}=\frac{1}{2} \\ & P\left(E_{2} \mid A\right)=\frac{P\left(E_{2}\right) P\left(A \mid E_{2}\right)}{P\left(E_{1}\right) P\left(A \mid E_{1}\right)+P\left(E_{2}\right) P\left(A \mid E_{2}\right)} \\ &=\frac{\dfrac{1}{2} \times \dfrac{1}{2}}{\dfrac{1}{2} \times \dfrac{1}{3}+\dfrac{1}{2} \times \dfrac{1}{2}} \end{aligned} \]