Let the events be defined as:

**A**: Selection of box A

**B**: Selection of box B

**R:** Drawing a red ball.

Let P (B) = p. Then, according to given condition P(A) = 2P (B) = 2p

\(P(\frac{R}{A}) =\frac{^3C_1}{^5C_1}= \frac{3}{5}\) ,\(P(\frac{R}{B}) =\frac{^4C_1}{^7C_1}= \frac{4}{7}\)

∴ \(P(\frac{B}{R}) = \frac{P(B).P(\frac{R}{B})}{P(A).P(\frac{R}{A})+P(B).P(\frac{R}{B})}\) = \(\frac{p.\frac{4}{7}} {2p.\frac{3}{5}+p.\frac{4}{7}}\) =\(\frac{4/7}{6/5+4/7}\) = \(\frac{4/7}{\frac{42+20}{35}} =\frac{4/7}{62/35}\)

= **\(\frac{10}{31}\)**