At least one color missing when drawing three balls

Question

An urn holds 9 balls: 3 red, 3 green, and 3 blue. Three balls are drawn at random without replacement. What is the probability that at least one color is completely missing from the draw?

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Let Ac be the event that color c is absent, and apply inclusion-exclusion over the three colors. Counts are over \binom{9}{3} = 84 equally likely draws. - One color absent: the three balls come from the other 6, so \binom{6}{3} = 20, with \binom{3}{1} = 3 colors. - Two colors absent: all three balls are one color, so \binom{3}{3} = 1, with \binom{3}{2} = 3 pairs. - All three colors absent is impossible. Alternating signs: 3(20) - 3(1) = 60 - 3 = 57. So P(	ext{some color missing}) = \frac{57}{84} = \frac{19}{28} \approx 0.6786. As a check, the complement (all three colors present) means one o…

At least one color missing when drawing three balls

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