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

Options

A. `(1)/(6)`
B. `(1)/(3)`
C. `(2)/(15)`
D. none of these

Explanation

To solve the problem of finding the probability that both marbles picked from the urn are red, we can follow these steps:

### Step 1: Determine the total number of marbles
First, we need to calculate the total number of marbles in the urn.

- Red marbles: 6
- Blue marbles: 4
- Green marbles: 2
- Yellow marbles: 3

**Total marbles = 6 (red) + 4 (blue) + 2 (green) + 3 (yellow) = 15 marbles**

### Step 2: Calculate the total number of ways to pick 2 marbles from 15
The total number of ways to choose 2 marbles from 15 can be calculated using the combination formula:

\[ \text{Total ways to choose 2 from 15} = \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 \]

### Step 3: Calculate the number of ways to pick 2 red marbles
Now, we need to find the number of ways to choose 2 red marbles from the 6 red marbles available:

\[ \text{Ways to choose 2 red from 6} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Step 4: Calculate the probability that both marbles are red
The probability \( P \) that both marbles picked are red can be calculated by dividing the number of favorable outcomes (picking 2 red marbles) by the total number of outcomes (picking any 2 marbles):

\[ P(\text{both are red}) = \frac{\text{Number of ways to choose 2 red}}{\text{Total ways to choose 2 from 15}} = \frac{15}{105} \]

### Step 5: Simplify the probability
Now, we simplify the fraction:

\[ P(\text{both are red}) = \frac{15}{105} = \frac{1}{7} \]

### Final Answer
The probability that both marbles picked are red is \( \frac{1}{7} \).

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