Question

There are three urns A, B, and C. Urn A contains 4 red balls and 3 black balls. urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that 3 balls drawn consists of 2 red balls and a black ball?

Answer 1

$$\begin{gathered} \text{Given:} \\ \text{Urn A}\left(4R+3B\right) \\ \text{Urn B}\left(5R+4B\right) \\ \text{Urn C}\left(4R+4B\right) \\ P\left(\text{two red and one black}\right)=P\left(\text{black from urn A}\right)+P\left(\text{black from urn B}\right)+P\left(\text{black from urn C}\right) \\ =\frac{3}{7}\times \frac{5}{9}\times \frac{4}{8}+\frac{4}{7}\times \frac{4}{9}\times \frac{4}{8}+\frac{4}{7}\times \frac{5}{9}\times \frac{4}{8} \\ =\frac{5}{42}\times \frac{16}{126}\times \frac{20}{126} \\ =\frac{15+16+20}{126} \\ =\frac{51}{126}=\frac{17}{42} \end{gathered}$$