Question

an urn contains 6 blue, 2 yellow, and 2 gray marbles. two marbles are randomly drawn from the urn. find the probability the two marbles are the same color.

$\frac{11}{25}$
$\frac{22}{45}$
$\frac{17}{45}$
$\frac{17}{50}$

Explanation:

Step1: Total marbles and combinations

Total marbles: \(6 + 2 + 2 = 10\). Total ways to draw 2: \( \binom{10}{2} = \frac{10 \times 9}{2} = 45 \).

Step2: Blue same color ways

Blue marbles: 6. Ways: \( \binom{6}{2} = \frac{6 \times 5}{2} = 15 \).

Step3: Yellow same color ways

Yellow marbles: 2. Ways: \( \binom{2}{2} = 1 \).

Step4: Gray same color ways

Gray marbles: 2. Ways: \( \binom{2}{2} = 1 \).

Step5: Favorable outcomes and probability

Favorable: \(15 + 1 + 1 = 17\). Probability: \( \frac{17}{45} \).

Answer:

C. \( \frac{17}{45} \)