An urn contains 4 black and 6 red balls. If two balls are drawn at random from the urn without replacement ,then the probability that both are black is

Options

A. `1/5`
B. `1/(15)`
C. `2/(15)`
D. `(13)/(15)`

Answer

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

### Step 1: Identify the total number of balls
The urn contains:
- 4 black balls
- 6 red balls

So, the total number of balls in the urn is:
\[ 4 + 6 = 10 \]

### Step 2: Calculate the total number of ways to draw 2 balls from 10
We need to calculate the total number of ways to choose 2 balls from the 10 balls. This can be calculated using the combination formula \( nCr \), which is given by:
\[ nCr = \frac{n!}{r!(n-r)!} \]
Here, \( n = 10 \) and \( r = 2 \):
\[ 10C2 = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

### Step 3: Calculate the number of ways to draw 2 black balls
Next, we need to calculate the number of ways to choose 2 black balls from the 4 black balls:
\[ 4C2 = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Calculate the probability
The probability of drawing 2 black balls can be calculated using the formula:
\[ P(\text{both black}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Substituting the values we calculated:
\[ P(\text{both black}) = \frac{4C2}{10C2} = \frac{6}{45} \]

### Step 5: Simplify the probability
Now we simplify \( \frac{6}{45} \):
\[ \frac{6}{45} = \frac{2}{15} \]

Thus, the probability that both balls drawn are black is:
\[ \frac{2}{15} \]

### Summary of Steps:
1. Count the total number of balls: 10.
2. Calculate total ways to choose 2 balls: 45.
3. Calculate ways to choose 2 black balls: 6.
4. Use the probability formula: \( \frac{6}{45} \).
5. Simplify to get \( \frac{2}{15} \).