Question

A bag contains 4 red and 6 blue balls. Two balls are drawn at random without replacement. What is the probability that both are red?

• A. 1/15
• B. 2/15
• C. 4/45
• D. 2/45

Hint:

Use combinations for probability without replacement, or verify with sequential probabilities, adjusting the total after each draw.

Answer 1

The Correct Option is B

To determine the probability of selecting two red balls without replacement, the combination method is applied:

1. Total balls: 4 red + 6 blue = 10 balls.
2. Calculate combinations of 2 red balls: \[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6. \]
3. Calculate total combinations of 2 balls from 10: \[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45. \]
4. Calculate probability: \[ P = \frac{\binom{4}{2}}{\binom{10}{2}} = \frac{6}{45} = \frac{2}{15}. \]
5. Alternative approach (sequential probability): \[ P(\text{first red}) = \frac{4}{10} = \frac{2}{5}, \] \[ P(\text{second red} \mid \text{first red}) = \frac{3}{9} = \frac{1}{3}, \] \[ P = \frac{2}{5} \times \frac{1}{3} = \frac{2}{15}. \]
6. Compare with options: \( \frac{2}{15} \approx 0.133 \).

Therefore, the correct answer is: \[ \boxed{\frac{2}{15}} \]