A bag contains 5 red, 4 blue, and 3 green balls. Three balls are drawn at random. What is the probability that at least two balls are of the same color?
Show Hint
- \( \frac{7}{11} \)
- \( \frac{8}{11} \)
- \( \frac{9}{11} \)
- \( \frac{10}{11} \)
The Correct Option is B
Solution and Explanation
Step 1: Calculate Total Ways to Draw 3 Balls
Total balls = 5 (red) + 4 (blue) + 3 (green) = 12. We need to choose 3 balls: \[ \text{Total ways} = \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
Step 2: Calculate Ways to Get All Different Colors
Choose 1 red, 1 blue, and 1 green: \[ \text{Ways} = 5 \times 4 \times 3 = 60 \]
Step 3: Calculate Probability of All Different Colors
\[ P(\text{all different}) = \frac{\text{Ways to get all different}}{\text{Total ways}} = \frac{60}{220} = \frac{3}{11} \]
Step 4: Calculate Probability of At Least Two Same
\[ P(\text{at least two same}) = 1 - P(\text{all different}) = 1 - \frac{3}{11} = \frac{8}{11} \]
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