A bag contains 5 red, 3 black balls and another   bag contains 4 red and 5 black balls. One of the bags is chosen at random and a draw of two balls is made from it. The chance that one is red and the other is black is

$\begin{array}{l}\mathrm{One} \mathrm{bag} \mathrm{is} \mathrm{selected} \mathrm{at} \mathrm{random}\\ p\left(A\right)=\frac{1}{2}p\left(B\right)=\frac{1}{2}\\ 2\mathrm{balls} \mathrm{are} \mathrm{drawn} \mathrm{from} \mathrm{a} \mathrm{selected} \mathrm{bag}\\ \mathrm{E}=\mathrm{drawing} \mathrm{one} \mathrm{red} \mathrm{and} \mathrm{one} \mathrm{black} \mathrm{from} \mathrm{the} \mathrm{selected} \mathrm{bag}.\\ P(E/A)=\frac{5c_1.3c_1}{8c_2},P(E/B)=\frac{4c_1.5c_1}{9c_2}\\ Required Probability=P\left(E\right)=P\left(A\right)P(E/A)+P\left(B\right)P(E/B)\\ \\ =\frac{1}{2}.\frac{5c_1.3c_1}{8c_2}+\frac{1}{2}\frac{4c_1.5c_1}{9c_2}\\ =\frac{1}{2}.\frac{5.3}{\frac{8\times 7}{1\times 2}}+\frac{1}{2}.\frac{4\times 5}{\frac{9\times 8}{1\times 2}}\\ =\frac{15}{56}+\frac{5}{18}=\frac{275}{504}\end{array}$