A lot of 4 white and 4 red balls is randomly divided into two halves. What is the probability that there will be 2 red and 2 white balls in each half ?

Options

A. `18//35`
B. `3//35`
C. `1//2`
D. None of these

Explanation

To solve the problem of finding the probability that when dividing 4 white and 4 red balls into two halves, each half contains 2 red and 2 white balls, we can follow these steps:

### Step 1: Understand the total number of balls
We have a total of 8 balls, consisting of 4 white and 4 red balls.

### Step 2: Calculate the total ways to choose 4 balls from 8
The total number of ways to choose 4 balls from 8 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose.

\[ \text{Total ways} = \binom{8}{4} \]

### Step 3: Calculate the number of favorable outcomes
We need to find the number of ways to choose 2 white balls from 4 and 2 red balls from 4.

- The number of ways to choose 2 white balls from 4 is:

\[ \text{Ways to choose white} = \binom{4}{2} \]

- The number of ways to choose 2 red balls from 4 is:

\[ \text{Ways to choose red} = \binom{4}{2} \]

### Step 4: Combine the favorable outcomes
The total number of favorable outcomes (2 white and 2 red) is the product of the two combinations calculated above:

\[ \text{Favorable outcomes} = \binom{4}{2} \times \binom{4}{2} \]

### Step 5: Calculate the probability
The probability \( P \) that each half will have 2 red and 2 white balls is given by the ratio of the number of favorable outcomes to the total number of ways to choose 4 balls from 8:

\[ P = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{\binom{4}{2} \times \binom{4}{2}}{\binom{8}{4}} \]

### Step 6: Calculate the values
Now we can calculate the values:

1. Calculate \( \binom{4}{2} \):

\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

2. Calculate \( \binom{8}{4} \):

\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

3. Substitute these values back into the probability formula:

\[ P = \frac{6 \times 6}{70} = \frac{36}{70} = \frac{18}{35} \]

### Final Answer
The probability that there will be 2 red and 2 white balls in each half is:

\[ \boxed{\frac{18}{35}} \]

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