Assume a box contains 2 red balls and 2 black balls. One black ball has been drawn and not replaced. Which statement best describes the distribution of the three-draw sequences from the set {R,R,B} without replacement?

Assume a box contains 2 red balls and 2 black balls. One black ball has been drawn and not replaced. Which statement best describes the distribution of the three-draw sequences from the set {R,R,B} without replacement?

• A. All three patterns are equally likely
• B. RRB is twice as likely as RBR
• C. RBR is most likely
• D. BRR is impossible

Answer: (A)

The key idea is that after drawing one black ball and not replacing it, the remaining set is two red and one black. Drawing all three without replacement is just listing all possible orders of these three balls. If you pretend the two red balls are distinct (R1 and R2), there are six equally likely orders of R1, R2, and B. Each color pattern (RRB, RBR, BRR) corresponds to exactly two of these six orders, so each pattern occurs with probability 2/6 = 1/3. Therefore, the three possible sequences—RRB, RBR, and BRR—are equally likely.

The key idea is that after drawing one black ball and not replacing it, the remaining set is two red and one black. Drawing all three without replacement is just listing all possible orders of these three balls. If you pretend the two red balls are distinct (R1 and R2), there are six equally likely orders of R1, R2, and B. Each color pattern (RRB, RBR, BRR) corresponds to exactly two of these six orders, so each pattern occurs with probability 2/6 = 1/3. Therefore, the three possible sequences—RRB, RBR, and BRR—are equally likely.