Single correct option
A bag contains 5 red, 4 blue and 3 green balls. Two balls are drawn at random without replacement. What is the probability that both balls drawn are of the same colour?
- A. 19/66
- B. 1/3
- C. 9/22
- D. 2/11
Show answer and solution
Answer: (A) 19/66
Total balls = 12, so the total ways to draw 2 = C(12, 2) = 66.
Favourable (same colour): C(5,2) + C(4,2) + C(3,2) = 10 + 6 + 3 = 19.
Probability = 19/66.
Why others fail: 1/3 (22/66) and 9/22 (27/66) overcount by treating the colours as equally likely pairs; 2/11 (12/66) undercounts by omitting one colour's pairs.
Tip: For 'both same colour' with unequal group sizes, always add C(nᵢ, 2) for each colour separately over the total C(N, 2) — don't average the individual probabilities.
Favourable (same colour): C(5,2) + C(4,2) + C(3,2) = 10 + 6 + 3 = 19.
Probability = 19/66.
Why others fail: 1/3 (22/66) and 9/22 (27/66) overcount by treating the colours as equally likely pairs; 2/11 (12/66) undercounts by omitting one colour's pairs.
Tip: For 'both same colour' with unequal group sizes, always add C(nᵢ, 2) for each colour separately over the total C(N, 2) — don't average the individual probabilities.
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