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 solve the problem of finding the probability that both balls drawn from a bag containing 5 black and 4 red balls are red, we can follow these steps:

### Step 1: Determine the total number of balls.
We have:
- Black balls = 5
- Red balls = 4

Total number of balls = 5 + 4 = 9

### Step 2: Calculate the total ways to draw 2 balls from 9.
The total number of ways to choose 2 balls from 9 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, \( n = 9 \) and \( r = 2 \):
\[ \text{Total ways} = \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

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

### Step 4: Calculate the probability that both balls are red.
The probability \( P \) that both balls drawn are red is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{both balls are red}) = \frac{\text{Ways to choose 2 red balls}}{\text{Total ways to choose 2 balls}} = \frac{6}{36} = \frac{1}{6} \]

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

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