\(P(A/E_1\))=\(P(getting\;black\;ball\;/1\;bag\;is\;chosen)\)

=\(\Large\frac{3}{7}\)

\(P(A/E_2\))=\(P(getting\;black\;ball\;/1\;bag\;is\;chosen)\)

=\(\Large\frac{4}{7}\)

\(P(A)=P(getting\;black\;ball\))

=\(\Large\frac{1}{3}\)\(\times\)\(\frac{3}{7}\)+\(\frac{2}{3}\)\(\times\)\(\frac{4}{7}\)

=\(\Large\frac{11}{21}\)