There are 5 black and 4 red balls ina bag. Two balls are drawn at random. Find the probability that both balls are red.

Explanation

To find the probability that both balls drawn from a bag containing 5 black balls and 4 red balls are red, we can follow these steps:

### Step 1: Determine the total number of balls
The total number of balls in the bag is:
- Black balls = 5
- Red balls = 4
- Total balls = 5 + 4 = 9

### Step 2: Calculate the total number of ways to choose 2 balls from 9
We can use the combination formula \( nCr \) to find the number of ways to choose 2 balls from 9:
\[ \text{Total ways} = \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 3: Calculate the number of ways to choose 2 red balls from 4
Next, we calculate the number of ways to choose 2 red balls from the 4 available:
\[ \text{Favorable ways} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Calculate the probability
The probability \( P \) that both balls drawn are red is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{both red}) = \frac{\text{Number of favorable ways}}{\text{Total ways}} = \frac{6}{36} = \frac{1}{6} \]

### Final Answer
Thus, the probability that both balls drawn are red is \( \frac{1}{6} \).

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