An urn contains 3 red and 4 green marbles (7 total). If three marbles are picked at random without replacement, what is the probability that two are green and one is red?

Introduction / Context:
Sampling without replacement from colored marbles is modeled by combinations. We count favorable color-compositions and divide by total equally likely 3-combinations from the urn.

Given Data / Assumptions:

• R = 3 (red), G = 4 (green), total N = 7.
• Draw k = 3 without replacement.
• Target composition: 2 green, 1 red.

Concept / Approach:

• Total ways = C(7,3).
• Favorable = C(4,2) * C(3,1).
• Probability = favorable / total.

Step-by-Step Solution:

Verification / Alternative check:
Hypergeometric model: HG(N=7, success-type G=4, draws k=3, want x=2) gives the same ratio 18/35 after multiplying by C(3,1) for the red pick.

Why Other Options Are Wrong:

• 3/7, 5/14, 4/21 result from mixing replacement assumptions or miscounting combinations.

Common Pitfalls:

• Treating order as distinct (it is not; combinations handle orderless draws).

Final Answer:
18/35