An urn contains 3 red and 4 green marbles. If three marbles are picked at random, what is the probability that two are green and one is red?

Options

A. `3/7`
B. `18/35`
C. `5/14`
D. `4/21`

Answer

To solve the problem of finding the probability that two marbles are green and one marble is red when picking three marbles from an urn containing 3 red and 4 green marbles, we can follow these steps:

### Step 1: Identify the total number of marbles
The urn contains:
- 3 red marbles
- 4 green marbles

Total number of marbles = 3 (red) + 4 (green) = 7 marbles.

### Step 2: Determine the favorable outcomes
We want to find the number of ways to pick 2 green marbles and 1 red marble.

- The number of ways to choose 2 green marbles from 4 green marbles is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Thus, we calculate:
\[ \text{Ways to choose 2 green marbles} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

- The number of ways to choose 1 red marble from 3 red marbles is:
\[ \text{Ways to choose 1 red marble} = \binom{3}{1} = \frac{3!}{1!(3-1)!} = 3 \]

### Step 3: Calculate the total favorable outcomes
Now, we multiply the number of ways to choose the green marbles by the number of ways to choose the red marble:
\[ \text{Total favorable outcomes} = \binom{4}{2} \times \binom{3}{1} = 6 \times 3 = 18 \]

### Step 4: Calculate the total outcomes when picking 3 marbles
Next, we need to find the total number of ways to choose any 3 marbles from the 7 marbles:
\[ \text{Total outcomes} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 5: Calculate the probability
Finally, we can calculate the probability of picking 2 green and 1 red marble:
\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{18}{35} \]

### Final Answer
The probability that two marbles are green and one marble is red is \( \frac{18}{35} \).

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