Question

A bag contains 5 red, 3 blue, and 2 green balls. If two balls are drawn at random without replacement, what is the probability that both are red?

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{5}{9} \)
• D. \( \frac{1}{6} \)

Hint:

Use combinations to calculate probabilities when dealing with random draws without replacement.

Answer 1

The Correct Option is D

The total number of balls is: \[ 5 + 3 + 2 = 10 \text{ balls}. \] The number of ways to draw 2 balls from 10 is: \[ \binom{10}{2} = \frac{10 \times 9}{2} = 45. \] The number of ways to draw 2 red balls from 5 red balls is: \[ \binom{5}{2} = \frac{5 \times 4}{2} = 10. \] Thus, the probability of drawing two red balls is: \[ \frac{10}{45} = \frac{1}{6}. \] The correct answer is: \[ \boxed{\frac{1}{6}}. \]