A basket contains 4 red , 5 blue and 3 green marbles .
If two marbles are picked at random , what is the probability that both are red ?

Options

A. `(3)/(7)`
B. `(1)/(2)`
C. `(2)/(11)`
D. none of these

Explanation

To find the probability that both marbles drawn from the basket are red, we can follow these steps:

### Step 1: Determine the total number of marbles.
The basket contains:
- 4 red marbles
- 5 blue marbles
- 3 green marbles

Total number of marbles = 4 (red) + 5 (blue) + 3 (green) = 12 marbles.

### Step 2: Calculate the total number of ways to pick 2 marbles from 12.
The total number of ways to choose 2 marbles from 12 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Here, \( n = 12 \) and \( r = 2 \):
\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66. \]

### Step 3: Calculate the number of ways to pick 2 red marbles from 4.
Now, we need to find the number of ways to choose 2 red marbles from the 4 red marbles:
\[ \text{Ways to choose 2 red marbles} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6. \]

### Step 4: Calculate the probability that both marbles are red.
The probability \( P \) that both marbles drawn are red is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{both red}) = \frac{\text{Number of ways to choose 2 red marbles}}{\text{Total ways to choose 2 marbles}} = \frac{6}{66} = \frac{1}{11}. \]

### Final Answer:
The probability that both marbles picked are red is \( \frac{1}{11} \).

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