Probability

Concept:Probability of sequential events under with replacement (independent draws) and without replacement (dependent draws).Explanation:Total balls = 5 white + 7 black = 12.Choose 3 balls in each draw.With replacement:$P(\text{first draw all white}) = \frac{\binom{5}{3}}{\binom{12}{3}} = \frac{10}{220}$$P(\text{second draw all black}) = \frac{\binom{7}{3}}{\binom{12}{3}} = \frac{35}{220}$Product = $\frac{10 \times 35}{220 \times 220} = \frac{350}{48400} = \frac{7}{968}$Without replacement:After first draw (3 white removed), remaining: 2 white, 7 black, total 9.$P(\text{first draw all white}) = \frac{10}{220}$$P(\text{second draw all black} \mid \text{first all white}) = \frac{\binom{7}{3}}{\binom{9}{3}} = \frac{35}{84}$Product = $\frac{10}{220} \times \frac{35}{84} = \frac{350}{18480} = \frac{5}{264}$Answer:The probabilities are $\frac{7}{968}$ and $\frac{5}{264}$ respectively, which corresponds to option D.