There are 5 white and 4 red balls in a bag. Two balls are drawn at random. Find the probability that both balls are white.

Explanation

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

### Step 1: Identify the Total Number of Balls
We have:
- 5 white balls
- 4 red balls

**Total number of balls = 5 + 4 = 9 balls**

### Step 2: Calculate the Total Number of Ways to Draw 2 Balls
To find the total number of ways to draw 2 balls from 9, we use the combination formula \( C(n, r) \), which is given by:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
Here, \( n = 9 \) and \( r = 2 \):
\[ C(9, 2) = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 3: Calculate the Number of Ways to Draw 2 White Balls
Next, we calculate the number of ways to draw 2 balls from the 5 white balls:
\[ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Calculate the Probability
The probability \( P \) that both balls drawn are white is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{both balls are white}) = \frac{\text{Number of ways to choose 2 white balls}}{\text{Total number of ways to choose 2 balls}} = \frac{C(5, 2)}{C(9, 2)} = \frac{10}{36} \]

### Step 5: Simplify the Probability
Now we simplify \( \frac{10}{36} \):
\[ \frac{10}{36} = \frac{5}{18} \]

### Final Answer
Thus, the probability that both balls drawn are white is:
\[ \frac{5}{18} \]

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