Probability

The probability of transferring a white ball from $A$ to $B=\frac{4}{5}$The probability of transferring a black ball from $A$ to $B=\frac{1}{5}$The probability of drawing a black ball from $C$ after transferring a white ball from $A$ to $B$$= \frac{4}{6} \cdot \frac{3}{6} + \frac{2}{6} \cdot \frac{4}{6} = \frac{12}{36} + \frac{8}{36} = \frac{20}{36}$The probability of drawing a black ball from $C$ after transferring a black ball from $A$ to $B$$= \frac{3}{6} \cdot \frac{3}{6} + \frac{3}{6} \cdot \frac{4}{6} = \frac{9}{36} + \frac{12}{36} = \frac{21}{36}$The total probability of drawing a black ball from $C$ is$= \frac{4}{5} \cdot \frac{20}{36} + \frac{1}{5} \cdot \frac{21}{36} = \frac{101}{180}$