Question

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?

• A. \( \frac{7}{11} \)
• B. \( \frac{8}{11} \)
• C. \( \frac{9}{11} \)
• D. \( \frac{10}{11} \)

Hint:

Key Fact: When calculating "at least" probabilities, it’s often easier to compute the complement (all different) and subtract from 1.

Answer 1

The Correct Option is B

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
\[ P(\text{at least two same}) = 1 - P(\text{all different}) = 1 - \frac{3}{11} = \frac{8}{11} \]