Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a​ girl, but assume that the method has no​ effect, so the probability of a girl is 0.5. Assume that the groups consist of

20 couples. Complete parts​ (a) through​ (c) below.


Find the mean and the standard deviation for the numbers of girls in groups of

20 births.


Since we are given that each couple has a probability of 0.5 of having a girl, this scenario can be modeled by a **binomial distribution**.

The parameters of a binomial distribution are:

- \( n = 20 \) (the number of trials, or births, in this case)
- \( p = 0.5 \) (the probability of each trial resulting in a girl)

### Part (a): Mean
The mean of a binomial distribution is calculated as:
\[
\mu = n \cdot p
\]

Plugging in the values:
\[
\mu = 20 \cdot 0.5 = 10
\]

So, the mean number of girls is **10**.

### Part (b): Standard Deviation
The standard deviation for a binomial distribution is given by:
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]

Plugging in the values:
\[
\sigma = \sqrt{20 \cdot 0.5 \cdot (1 - 0.5)} = \sqrt{20 \cdot 0.5 \cdot 0.5} = \sqrt{5} \approx 2.24
\]

So, the standard deviation of the number of girls is approximately **2.24**.

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Let me know if you'd like more details or have any questions about these calculations. 

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Here are five follow-up questions you might explore:

1. What would the mean and standard deviation be if the probability of having a girl was 0.6 instead?
2. How would these values change if there were 50 couples instead of 20?
3. What is the probability of having exactly 10 girls in a group of 20 births?
4. How does the standard deviation help in understanding the distribution of outcomes?
5. What is the probability of having more than 15 girls in a group of 20 births?

**Tip:** In a binomial distribution, if the probability remains the same, increasing the number of trials increases both the mean and the variability (standard deviation) of the outcome.

Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a​ girl, but assume that the method has no​ effect, so the probability of a girl is 0.5. Assume that the groups consist of 20 couples. Complete parts​ (a) through​ (c) below. Find the mean and the standard deviation for the numbers of girls in groups of 20 births.

Learn how to calculate the mean and standard deviation for the number of girls in groups of 20 births using the binomial distribution. This problem covers probability concepts and formulas, ideal for students in Grades 10-12.

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