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.