Question

A box contains twelve balls of which 4 are red, 5 are green, and 3 are white. If three balls are drawn at random, the probability that exactly 2 balls have the same color is

• A. \( \frac{27}{44} \)
• B. \( \frac{29}{44} \)
• C. \( \frac{17}{22} \)
• D. \( \frac{31}{44} \)

Hint:

To find probabilities in problems with conditions on specific outcomes, first calculate the total number of possible outcomes, then count the favorable ones using combinations.

Answer 1

The Correct Option is B

The total number of ways to draw 3 balls from 12 is \( \binom{12}{3} = 220 \).
We need the probability of drawing exactly 2 balls of the same color.
Consider cases: - Case 1: 2 Red balls, 1 non-Red (Green or White).
\( \binom{4}{2} \times \binom{8}{1} = 6 \times 8 = 48 \).
- Case 2: 2 Green balls, 1 non-Green (Red or White).
\( \binom{5}{2} \times \binom{7}{1} = 10 \times 7 = 70 \).
- Case 3: 2 White balls, 1 non-White (Red or Green).
\( \binom{3}{2} \times \binom{9}{1} = 3 \times 9 = 27 \).
Total favorable outcomes: \( 48 + 70 + 27 = 145 \).
Probability = \( \frac{145}{220} = \frac{29}{44} \), matching option (2).