A bag contains 4 white, 5 black, and 6 red balls. Two balls are drawn randomly. What is the probability that they are of different colors?
Instruction: Select two distinct balls uniformly without replacement from four white, five black and six red balls. Use combinations to calculate the probability of different colors, and give a numerical result.
Context: This question examines the candidate's aptitude in combinatorial probability and understanding of without replacement scenario.
Updated
Official answer
Choose two distinct balls uniformly, without replacement. There are C(15,2) = 105 possible unordered pairs.
Different-color pairs consist of white/black, white/red or black/red. Their counts are 4×5 = 20, 4×6 = 24 and 5×6 = 30. These disjoint cases total 74, giving P(different colors) = 74/105, approximately 70.48%.
The complement gives the same result: 1 − [C(4,2)+C(5,2)+C(6,2)]/105 = 1−31/105. Do not multiply the unordered-pair count by two; both draw orders are already represented by the same pair.