Question

A town has 10,000 two-child families. Design a simulation to estimate the percentage of two-child families with two girls. Choose a simulation tool, explain one trial, and describe the headings for a table in which you would record the outcomes.

Answer 1

To estimate the percentage of two-child families with two girls using a simulation, we can use a simple randomization method. For the sake of this example, we'll use a coin-flipping simulation to represent the genders of the two children in each family. We can assume the coin has an equal chance of landing on heads (representing a girl) or tails (representing a boy).

### Step-by-Step Simulation Design:

1. **Simulation Tool**:
We will use a simple programming language like Python to simulate the scenarios, but this could also be done with physical coins, dice, or a spreadsheet with random number generation.

2. **Define One Trial**:
Each trial will represent one family with two children. To conduct one trial:
- Flip a coin twice:
- The first flip represents the first child's gender.
- The second flip represents the second child's gender.
- Record the outcome of each trial as follows:
- Heads (H) = Girl
- Tails (T) = Boy
- Count the combinations for each family (HH, HT, TH, TT).

### Example of One Trial:

Suppose we conduct one trial where the outcomes of the two flips are:
- First flip: Heads (Girl)
- Second flip: Tails (Boy)

Thus, the family structure for this trial would be (Girl, Boy).

### Table Headings for Outcomes:

We would record the results of our simulation in a table with the following headings:

| Trial Number | First Child | Second Child | Outcome |
|--------------|-------------|--------------|---------------------------|
| 1 | H | T | Girl, Boy |
| 2 | T | H | Boy, Girl |
| 3 | H | H | Girl, Girl (Two Girls) |
| 4 | T | T | Boy, Boy |
| ... | ... | ... | ... |
| 10,000 | H | H | Girl, Girl (Two Girls) |

### Running the Simulation:
- Repeat the trial 10,000 times to account for the total number of families.
- After all trials, count how many families have the outcome (Girl, Girl) i.e., HH.

### Calculating the Percentage:
- Calculate the percentage of families with two girls using the formula:
\[
\text{Percentage} = \left( \frac{\text{Number of HH outcomes}}{10,000} \right) \times 100
\]

This simulation will help estimate the percentage of two-child families with two girls accurately. As per probability theory, we expect approximately 25% of the outcomes to have two girls (1 out of 4 combinations). The simulation will provide a practical demonstration of this theory.