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The problem in the image states:

"The random variable \( x \) represents the number of girls in a family of three children. Assuming that boys and girls are equally likely, find the mean and standard deviation for the random variable \( x \). Round to two decimal places."

### Step 1: Define the problem.
- The number of trials (children) \( n = 3 \).
- The probability of having a girl \( p = 0.5 \) (since boys and girls are equally likely).
- The number of girls \( x \) follows a binomial distribution: \( x \sim Binomial(n = 3, p = 0.5) \).

### Step 2: Mean of a binomial distribution.
The mean \( \mu \) of a binomial distribution is given by:

\[
\mu = n \times p
\]

Substitute \( n = 3 \) and \( p = 0.5 \):

\[
\mu = 3 \times 0.5 = 1.5
\]

### Step 3: Standard deviation of a binomial distribution.
The formula for the standard deviation \( \sigma \) of a binomial distribution is:

\[
\sigma = \sqrt{n \times p \times (1 - p)}
\]

Substitute \( n = 3 \) and \( p = 0.5 \):

\[
\sigma = \sqrt{3 \times 0.5 \times (1 - 0.5)} = \sqrt{3 \times 0.5 \times 0.5} = \sqrt{0.75}
\]

\[
\sigma \approx 0.866
\]

### Final Answer:
- **Mean**: \( \mu = 1.50 \)
- **Standard Deviation**: \( \sigma \approx 0.87 \)

Would you like further details or clarification on this solution?

### Related Questions:
1. How would the mean and standard deviation change if the family had four children instead of three?
2. What is the probability of having exactly 2 girls in this scenario?
3. Can you calculate the variance for this binomial distribution?
4. How would the results change if the probability of having a girl was 0.6 instead of 0.5?
5. What is the probability of having no girls in a family of three children?

### Tip:
In binomial distributions, the variance \( \sigma^2 \) is simply \( n \times p \times (1 - p) \). This can help you quickly check or calculate deviations.

The random variable x represents the number of girls in a family of three children. Assuming that boys and girls are equally likely, find the mean and standard deviation for the random variable x. Round to two decimal places.

Calculate the mean and standard deviation for the number of girls in a family of three children. Using binomial distribution with a probability of 0.5 for both boys and girls, find the results rounded to two decimal places. Suitable for high school students.

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