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. `2//25`

Answer

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

### Step 1: Identify the Total Number of Ways to Choose Balls
We have a total of 8 balls (4 white and 4 red). We need to divide them into two groups of 4 balls each. The total number of ways to choose 4 balls from 8 is given by the combination formula:

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

### Step 2: Calculate the Total Ways to Choose 4 Balls
Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\):

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

### Step 3: Identify the Favorable Outcomes
Next, we need to find the number of ways to choose 2 red balls from 4 and 2 white balls from 4 for one half. This can be calculated as follows:

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

### Step 4: Calculate the Number of Favorable Outcomes
Calculating the combinations:

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

Thus, the total number of favorable outcomes (choosing 2 red and 2 white) is:

\[ \text{Favorable outcomes} = \binom{4}{2} \times \binom{4}{2} = 6 \times 6 = 36 \]

### Step 5: Calculate the Probability
Now, we can find the probability that each half will have 2 red and 2 white balls:

\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total ways}} = \frac{36}{70} \]

### Step 6: Simplify the Probability
To simplify \(\frac{36}{70}\):

\[ \frac{36}{70} = \frac{18}{35} \]

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

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

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