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 the bag are white, we can follow these steps:

### Step 1: Determine the total number of balls
In the bag, there are:
- 5 white balls
- 4 red balls

Total number of balls = 5 (white) + 4 (red) = 9 balls.

### Step 2: Calculate the total number of ways to draw 2 balls
The total number of ways to choose 2 balls from 9 can be calculated using the combination formula:
\[ \text{Total ways to choose 2 balls} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Where \( n \) is the total number of balls and \( r \) is the number of balls to choose.

So, we calculate:
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = 36 \]

### Step 3: Calculate the number of ways to draw 2 white balls
Now, we need to find the number of ways to choose 2 white balls from the 5 white balls available:
\[ \text{Ways to choose 2 white balls} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

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

### Step 5: Simplify the probability
Now we simplify the fraction:
\[ P(\text{both balls are white}) = \frac{10}{36} = \frac{5}{18} \]

### Final Answer
The probability that both balls drawn are white is \(\frac{5}{18}\).

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