construct a probability mass function in a family of four siblings, let G be the number of girls in the family



To construct the probability mass function (PMF) for the random variable \( G \), which represents the number of girls in a family of four siblings, we'll make the following assumptions:

1. **Binary Gender Model:** Each child can be either a girl (G) or a boy (B).
2. **Independence:** The gender of each child is independent of the others.
3. **Equal Probability:** The probability of having a girl \( P(G) = 0.5 \) and the probability of having a boy \( P(B) = 0.5 \).

### Possible Values for \( G \)
Since the family has four children, the number of girls \( G \) can take on the following values: 0, 1, 2, 3, or 4.

### Binomial Distribution
The number of girls \( G \) follows a binomial distribution:

\[
P(G = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( n = 4 \) (total number of children),
- \( k \) is the number of girls (ranging from 0 to 4),
- \( p = 0.5 \) (probability of having a girl).

### Calculating the PMF

1. **\( P(G = 0) \)**: Probability of having 0 girls (all boys):
   \[
   P(G = 0) = \binom{4}{0} (0.5)^0 (0.5)^4 = 1 \times 1 \times \frac{1}{16} = \frac{1}{16}
   \]

2. **\( P(G = 1) \)**: Probability of having 1 girl:
   \[
   P(G = 1) = \binom{4}{1} (0.5)^1 (0.5)^3 = 4 \times \frac{1}{2} \times \frac{1}{8} = \frac{4}{16} = \frac{1}{4}
   \]

3. **\( P(G = 2) \)**: Probability of having 2 girls:
   \[
   P(G = 2) = \binom{4}{2} (0.5)^2 (0.5)^2 = 6 \times \frac{1}{4} \times \frac{1}{4} = \frac{6}{16} = \frac{3}{8}
   \]

4. **\( P(G = 3) \)**: Probability of having 3 girls:
   \[
   P(G = 3) = \binom{4}{3} (0.5)^3 (0.5)^1 = 4 \times \frac{1}{8} \times \frac{1}{2} = \frac{4}{16} = \frac{1}{4}
   \]

5. **\( P(G = 4) \)**: Probability of having 4 girls (no boys):
   \[
   P(G = 4) = \binom{4}{4} (0.5)^4 (0.5)^0 = 1 \times \frac{1}{16} \times 1 = \frac{1}{16}
   \]

### Final PMF Table

| \( G \) | \( P(G = k) \)   |
|:------:|:----------------:|
|   0    | \( \frac{1}{16} \) |
|   1    | \( \frac{1}{4} \)  |
|   2    | \( \frac{3}{8} \)  |
|   3    | \( \frac{1}{4} \)  |
|   4    | \( \frac{1}{16} \) |

This table represents the probability mass function (PMF) for the number of girls in a family of four children.

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Would you like more details or have any other questions? Here are some related questions you might consider:

1. How would the PMF change if the probability of having a girl were different from 0.5?
2. What is the expected value (mean) of the number of girls in this family?
3. How do we calculate the variance for the number of girls in this family?
4. How does this binomial distribution compare to other distributions like the Poisson distribution?
5. Can you apply the same method to find the PMF for the number of boys?

**Tip:** When dealing with binomial distributions, remember that the sum of all probabilities in the PMF must equal 1, which helps verify your calculations.

Construct a probability mass function in a family of four siblings, let G be the number of girls in the family.

Explore how to construct a probability mass function (PMF) for the number of girls in a family of four siblings. This problem involves key concepts in probability and binomial distribution, and includes detailed calculations using the binomial coefficient. Suitable for students in Grades 9-12.

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