Threre are 9 red,7 white and 4 black balls in an urn.
Two balls are drawn at random from the urn. What will be the probability that both the balls are red ?

Options

A. `(18)/(95)`
B. `(9)/(95)`
C. `(9)/(10)`
D. `(8)/(95)`

Explanation

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

### Step 1: Determine the total number of balls in the urn.
We have:
- 9 red balls
- 7 white balls
- 4 black balls

**Total number of balls = 9 + 7 + 4 = 20 balls.**

### Step 2: Calculate the total number of ways to draw 2 balls from 20.
The number of ways to choose 2 balls from 20 is given by the combination formula \( nCr \), which is calculated as:

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

For our case:

\[ 20C2 = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190 \]

### Step 3: Calculate the number of ways to draw 2 red balls from the 9 red balls.
Using the combination formula again:

\[ 9C2 = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 4: Calculate the probability that both balls drawn are red.
The probability \( P \) is given by the ratio of the number of favorable outcomes (drawing 2 red balls) to the total number of outcomes (drawing any 2 balls):

\[ P(\text{both balls are red}) = \frac{\text{Number of ways to choose 2 red balls}}{\text{Total ways to choose 2 balls}} = \frac{36}{190} \]

### Step 5: Simplify the probability.
We can simplify \( \frac{36}{190} \):

\[ \frac{36}{190} = \frac{18}{95} \]

Thus, the probability that both balls drawn are red is:

\[ \boxed{\frac{18}{95}} \]

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