Question

A bag contains 4 red, 3 white, and 2 blue balls. Two balls are drawn at random. What is the probability that both are red?

• A. \( \frac{1}{6} \)
• B. \( \frac{1}{12} \)
• C. \( \frac{1}{9} \)
• D. \( \frac{2}{9} \)

Hint:

Use combinations \( \frac{\binom{\text{favorable}}{\text{total}}} \) for probability of selecting specific items.

Answer 1

The Correct Option is A

We need the probability of drawing two red balls.
- Step 1: Determine total balls. Total balls = \( 4 + 3 + 2 = 9 \).
- Step 2: Calculate total ways to draw 2 balls. Use combinations:
\[ \binom{9}{2} = \frac{9 \times 8}{2} = 36 \] - Step 3: Calculate favorable ways. Red balls = 4. Ways to draw 2 red balls:
\[ \binom{4}{2} = \frac{4 \times 3}{2} = 6 \] - Step 4: Compute probability. Probability = favorable ways / total ways:
\[ \frac{\binom{4}{2}}{\binom{9}{2}} = \frac{6}{36} = \frac{1}{6} \] - Step 5: Verify calculation. Alternative approach (sequential drawing):
- First red: \( \frac{4}{9} \). Second red: \( \frac{3}{8} \). Probability = \( \frac{4}{9} \times \frac{3}{8} = \frac{12}{72} = \frac{1}{6} \).
- Step 6: Check options.
- (a) \( \frac{1}{6} \): Correct.
- (b) \( \frac{1}{12} \): Incorrect.
- (c) \( \frac{1}{9} \): Incorrect.
- (d) \( \frac{2}{9} \): Incorrect.
Thus, the answer is a.