Answer: (c) \(\frac{10}{19}\)
Let the events E1, E2, and A be defined as follows:
E1 = Choosing bag A
E2 = Choosing bag B
A = Choosing red ball.
Then, P(E1) = P(E2) = \(\frac{1}{2}\)
(∵ There are two bags that have an equally likely chance of being chosen)
P(A/E1) = P(Drawing red ball from bag A) = \(\frac{2}{4} =\frac{1}{2}\) (∵ 2 red out of 4 balls)
P(A/E2) = P(Drawing red ball from bag B) = \(\frac{5}{9}\) (5 red out of 9 balls)
∴ P(Red balls are drawn from bag B)
\(=P(E_2/A) = \frac{P(E_2)\times P(A/E_2)}{P(E_1)\times P(A/E_1)+P(E_2)\times P(A/E_2)}\)
\(=\frac{\frac{1}{2}\times \frac{5}{9}}{\frac{1}{2}\times \frac{1}{2}+ \frac{1}{2}\times \frac{5}{9}}\) = \(\frac{\frac{5}{18}}{\frac{1}{4}+\frac{5}{18}}\) = \(=\frac{5/18}{19/36}\) = \(\frac{10}{19}\)