Anurn contains 6 red,4 blue ,2 green and 3yellow marbles
If two marbles are picked at random , what is the probability that either both are green or both are yellow?

Options

A. `(5)/(91)`
B. `(1)/(35)`
C. `(1)/(3)`
D. `(4)/(105)`

Explanation

To solve the problem of finding the probability that either both marbles picked are green or both are yellow, we can follow these steps:

### Step 1: Determine the total number of marbles
An urban contains:
- 6 red marbles
- 4 blue marbles
- 2 green marbles
- 3 yellow marbles

**Total number of marbles = 6 + 4 + 2 + 3 = 15**

### Step 2: Calculate the total possible outcomes for picking 2 marbles
The total number of ways to choose 2 marbles from 15 can be calculated using the combination formula \( C(n, r) \), which is given by:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

For our case, \( n = 15 \) and \( r = 2 \):

\[ C(15, 2) = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 \]

### Step 3: Calculate the favorable outcomes for both events
1. **Both marbles are green:**
There are 2 green marbles, so the number of ways to choose 2 green marbles is:

\[ C(2, 2) = 1 \]

2. **Both marbles are yellow:**
There are 3 yellow marbles, so the number of ways to choose 2 yellow marbles is:

\[ C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3 \]

### Step 4: Calculate the total favorable outcomes
Now, we add the favorable outcomes from both events:

\[ \text{Total favorable outcomes} = \text{(both green)} + \text{(both yellow)} = 1 + 3 = 4 \]

### Step 5: Calculate the probability
The probability \( P \) of either both marbles being green or both being yellow is given by the ratio of the total favorable outcomes to the total possible outcomes:

\[ P = \frac{\text{Total favorable outcomes}}{\text{Total possible outcomes}} = \frac{4}{105} \]

### Final Answer
The probability that either both marbles are green or both are yellow is \( \frac{4}{105} \).

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